Indices and surds
By the end of this chapter you will be able to
- Read a fractional index as a root and a power, and find its value
- Use negative indices to write reciprocals, and work out their values
- Simplify surds and add or subtract like surds
- Multiply and expand brackets that contain surds, and simplify the result
- Extension: rationalise a denominator with a single surd or a conjugate pair
You should already know
- Index notation and the index laws with whole-number powers (Year 8, Chapter 1)
- Square and cube numbers, and square roots (Year 8, Chapter 2)
- Simplifying a surd such as √50 = 5√2 and multiplying two brackets (Year 8, Chapters 2 and 5)
Where this leads
- Standard form and bounds (Year 9, Chapter 2)
- Quadratic expressions and the difference of two squares (Year 9, Chapter 4)
- Algebraic index laws with letters (GCSE)
National Curriculum for England, Key Stage 3 mathematics, Year 9. Content: Use fractional indices, such as 8(1/3) and 9(3/2), and move between root and index forms, Use negative and fractional index laws in calculations, Simplify surds, add and subtract like surds, and multiply and expand brackets containing surds, Extension: rationalise a denominator.
Exercise P Do you remember?
Non-calculatorMethod. Multiply the number by itself four times.
- Multiply the first two. Two 2s make 4.2 × 2 = 4
- Multiply by the next 2. Four 2s make 8.4 × 2 = 8
- Multiply by the last 2. This makes four 2s in total.8 × 2 = 16
Check. 24 = 16, so the answer is the fourth power of 2.
Method. Count the copies of 5 and write the index.
- Count the copies. There are three 5s.three copies of 5
- Write the index form. The base is 5 and the index is 3.53
Check. 53 = 125, which is 5 × 5 × 5.
Method. Find the number that multiplied by itself gives 81.
- Think of the square. We need a number times itself to make 81.9 × 9 = 81
- Write the root. The positive square root of 81 is 9.√81 = 9
Check. 9 × 9 = 81, so √81 = 9.
Method. Find the number that multiplied by itself three times gives 8.
- Try a number. Try 2. Two multiplied by itself three times gives 8.2 × 2 × 2 = 8
- Write the root. So the cube root of 8 is 2.2
Check. 23 = 8, so the cube root of 8 is 2.
Method. Find the square numbers either side of 50.
- List the square numbers near 50. Square numbers are 1, 4, 9, ..., 49, 64.72 = 49, 82 = 64
- Compare 50. 50 is not 49 and not 64, so it is not a square number.49 < 50 < 64
- Write the answer. No
Check. The square root of 50 is about 7.07, which is not a whole number.
Fractional indices as roots
A fractional index tells us to take a root. The index 1/2 means the square root, and 1/3 means the cube root. In general, a(1/n) is the n-th root of a.
Here is the idea with 8(1/3). We need a number that, multiplied by itself three times, gives 8. That number is 2, so 8(1/3) = 2.
Key ideas
- a(1/2) is the square root of a: 9(1/2) = √9 = 3.
- a(1/3) is the cube root of a: 27(1/3) = 3.
- The denominator of the index tells you which root to take.
- Brackets keep the fraction together in the index: 8(1/3).
Method. Work out the root from the denominator, then the power from the numerator. The index (1/4) is the fourth root of 16, with the power 1.
- Use the denominator as the root. The denominator 4 means the fourth root. The numerator is the power.16(1/4) = (fourth root of 16)1
- Find the root. 2 multiplied by itself 4 times gives 16, so 2 is the fourth root.2 × 2 × 2 × 2 = 16, so the fourth root of 16 is 2
- Raise the root to the power. The numerator is 1, so raise 2 to the power 1.21 = 2
- Write the answer. 2
Check. The fourth root of 16 is 2, and 21 = 2.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/3) is the cube root of 125, with the power 1.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.125(1/3) = (cube root of 125)1
- Find the root. 5 multiplied by itself 3 times gives 125, so 5 is the cube root.5 × 5 × 5 = 125, so the cube root of 125 is 5
- Raise the root to the power. The numerator is 1, so raise 5 to the power 1.51 = 5
- Write the answer. 5
Check. The cube root of 125 is 5, and 51 = 5.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. The cube root of a number is the same as raising it to the power 1/3.
- Read the root. This is the cube root of 64.3√64
- Write the root as a power. The cube root is the power 1/3.641/3 = 3√64
- Write the answer. Brackets keep the 1/n together.64(1/3)
Check. Raising 64(1/3) to the power 3 gives 64.
Watch out. Keep the 1/ in brackets. Without brackets 31/2 can be misread.
Watch out
Watch out: 8(1/3) is not 8 ÷ 3. The index tells you to find a root, so 8(1/3) = 2, not 2.67.
Exam tip
Exam tip. If you cannot see the root straight away, list the powers: 23 = 8, 33 = 27, 43 = 64.
Exercise 1A Fractional indices as roots
Non-calculatorMethod. Work out the root from the denominator, then the power from the numerator. The index (1/2) is the square root of 9, with the power 1.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.9(1/2) = (square root of 9)1
- Find the root. 3 multiplied by itself 2 times gives 9, so 3 is the square root.3 × 3 = 9, so the square root of 9 is 3
- Raise the root to the power. The numerator is 1, so raise 3 to the power 1.31 = 3
- Write the answer. 3
Check. The square root of 9 is 3, and 31 = 3.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/3) is the cube root of 27, with the power 1.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.27(1/3) = (cube root of 27)1
- Find the root. 3 multiplied by itself 3 times gives 27, so 3 is the cube root.3 × 3 × 3 = 27, so the cube root of 27 is 3
- Raise the root to the power. The numerator is 1, so raise 3 to the power 1.31 = 3
- Write the answer. 3
Check. The cube root of 27 is 3, and 31 = 3.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/3) is the cube root of 64, with the power 1.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.64(1/3) = (cube root of 64)1
- Find the root. 4 multiplied by itself 3 times gives 64, so 4 is the cube root.4 × 4 × 4 = 64, so the cube root of 64 is 4
- Raise the root to the power. The numerator is 1, so raise 4 to the power 1.41 = 4
- Write the answer. 4
Check. The cube root of 64 is 4, and 41 = 4.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/2) is the square root of 16, with the power 1.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.16(1/2) = (square root of 16)1
- Find the root. 4 multiplied by itself 2 times gives 16, so 4 is the square root.4 × 4 = 16, so the square root of 16 is 4
- Raise the root to the power. The numerator is 1, so raise 4 to the power 1.41 = 4
- Write the answer. 4
Check. The square root of 16 is 4, and 41 = 4.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/5) is the fifth root of 32, with the power 1.
- Use the denominator as the root. The denominator 5 means the fifth root. The numerator is the power.32(1/5) = (fifth root of 32)1
- Find the root. 2 multiplied by itself 5 times gives 32, so 2 is the fifth root.2 × 2 × 2 × 2 × 2 = 32, so the fifth root of 32 is 2
- Raise the root to the power. The numerator is 1, so raise 2 to the power 1.21 = 2
- Write the answer. 2
Check. The fifth root of 32 is 2, and 21 = 2.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/4) is the fourth root of 81, with the power 1.
- Use the denominator as the root. The denominator 4 means the fourth root. The numerator is the power.81(1/4) = (fourth root of 81)1
- Find the root. 3 multiplied by itself 4 times gives 81, so 3 is the fourth root.3 × 3 × 3 × 3 = 81, so the fourth root of 81 is 3
- Raise the root to the power. The numerator is 1, so raise 3 to the power 1.31 = 3
- Write the answer. 3
Check. The fourth root of 81 is 3, and 31 = 3.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/3) is the cube root of 1000, with the power 1.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.1000(1/3) = (cube root of 1000)1
- Find the root. 10 multiplied by itself 3 times gives 1000, so 10 is the cube root.10 × 10 × 10 = 1000, so the cube root of 1000 is 10
- Raise the root to the power. The numerator is 1, so raise 10 to the power 1.101 = 10
- Write the answer. 10
Check. The cube root of 1000 is 10, and 101 = 10.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/2) is the square root of 100, with the power 1.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.100(1/2) = (square root of 100)1
- Find the root. 10 multiplied by itself 2 times gives 100, so 10 is the square root.10 × 10 = 100, so the square root of 100 is 10
- Raise the root to the power. The numerator is 1, so raise 10 to the power 1.101 = 10
- Write the answer. 10
Check. The square root of 100 is 10, and 101 = 10.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/2) is the square root of 4, with the power 1.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.4(1/2) = (square root of 4)1
- Find the root. 2 multiplied by itself 2 times gives 4, so 2 is the square root.2 × 2 = 4, so the square root of 4 is 2
- Raise the root to the power. The numerator is 1, so raise 2 to the power 1.21 = 2
- Write the answer. 2
Check. The square root of 4 is 2, and 21 = 2.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/3) is the cube root of 8, with the power 1.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.8(1/3) = (cube root of 8)1
- Find the root. 2 multiplied by itself 3 times gives 8, so 2 is the cube root.2 × 2 × 2 = 8, so the cube root of 8 is 2
- Raise the root to the power. The numerator is 1, so raise 2 to the power 1.21 = 2
- Write the answer. 2
Check. The cube root of 8 is 2, and 21 = 2.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/4) is the fourth root of 16, with the power 1.
- Use the denominator as the root. The denominator 4 means the fourth root. The numerator is the power.16(1/4) = (fourth root of 16)1
- Find the root. 2 multiplied by itself 4 times gives 16, so 2 is the fourth root.2 × 2 × 2 × 2 = 16, so the fourth root of 16 is 2
- Raise the root to the power. The numerator is 1, so raise 2 to the power 1.21 = 2
- Write the answer. 2
Check. The fourth root of 16 is 2, and 21 = 2.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. The cube root of a number is the same as raising it to the power 1/3.
- Read the root. This is the cube root of 64.3√64
- Write the root as a power. The cube root is the power 1/3.641/3 = 3√64
- Write the answer. Brackets keep the 1/n together.64(1/3)
Check. Raising 64(1/3) to the power 3 gives 64.
Watch out. Keep the 1/ in brackets. Without brackets 31/2 can be misread.
Method. A square root is the same as the index 1/2.
- Read the square root. The square root of 7 is written with the square root sign.√7
- Use the index 1/2. A square root is the same as raising to the power 1/2.√7 = 7(1/2)
- Write the answer. 7(1/2)
Check. Squaring 7(1/2) gives 7, so it is the square root of 7.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/3) is the cube root of 216, with the power 1.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.216(1/3) = (cube root of 216)1
- Find the root. 6 multiplied by itself 3 times gives 216, so 6 is the cube root.6 × 6 × 6 = 216, so the cube root of 216 is 6
- Raise the root to the power. The numerator is 1, so raise 6 to the power 1.61 = 6
- Write the answer. 6
Check. The cube root of 216 is 6, and 61 = 6.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. The index 1/2 means take the square root.
- Read the index. The denominator 2 means the square root.64(1/2) = √64
- Find the square root. 8 × 8 = 64.√64 = 8
- Compare with 32. 32 is half of 64, so it is not the square root.8 ≠ 32
Check. 8 × 8 = 64, so 8 is the square root of 64.
Watch out. Do not halve the number for an index of 1/2. Take the square root instead.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/2) is the square root of 49, with the power 1.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.49(1/2) = (square root of 49)1
- Find the root. 7 multiplied by itself 2 times gives 49, so 7 is the square root.7 × 7 = 49, so the square root of 49 is 7
- Raise the root to the power. The numerator is 1, so raise 7 to the power 1.71 = 7
- Write the answer. 7
Check. The square root of 49 is 7, and 71 = 7.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the square root first, then multiply by 2.
- Work out 49(1/2). The square root of 49 is 7.49(1/2) = 7
- Multiply by 2. Multiply the answer by 2.7 × 2 = 14
Check. The square root of 49 is 7, and 7 × 2 = 14.
Fractional indices with a numerator
An index such as m/n has two parts. The denominator n tells you to take the n-th root. The numerator m tells you to raise the answer to the power m. Do the root first, because it keeps the numbers small.
So 8(2/3) means the cube root of 8, which is 2, and then squared: 22 = 4.
Key ideas
- a(m/n) = (n-th root of a)m. Find the root first, then the power.
- The denominator is the root. The numerator is the power.
- Working with the root first keeps the numbers small: 9(3/2) = 33 = 27.
- The answer for a positive index is always positive.
Method. Work out the root from the denominator, then the power from the numerator. The index (2/3) is the cube root of 8, with the power 2.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.8(2/3) = (cube root of 8)2
- Find the root. 2 multiplied by itself 3 times gives 8, so 2 is the cube root.2 × 2 × 2 = 8, so the cube root of 8 is 2
- Raise the root to the power. The numerator is 2, so raise 2 to the power 2.22 = 4
- Write the answer. 4
Check. The cube root of 8 is 2, and 22 = 4.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (3/2) is the square root of 9, with the power 3.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.9(3/2) = (square root of 9)3
- Find the root. 3 multiplied by itself 2 times gives 9, so 3 is the square root.3 × 3 = 9, so the square root of 9 is 3
- Raise the root to the power. The numerator is 3, so raise 3 to the power 3.33 = 27
- Write the answer. 27
Check. The square root of 9 is 3, and 33 = 27.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (3/4) is the fourth root of 16, with the power 3.
- Use the denominator as the root. The denominator 4 means the fourth root. The numerator is the power.16(3/4) = (fourth root of 16)3
- Find the root. 2 multiplied by itself 4 times gives 16, so 2 is the fourth root.2 × 2 × 2 × 2 = 16, so the fourth root of 16 is 2
- Raise the root to the power. The numerator is 3, so raise 2 to the power 3.23 = 8
- Write the answer. 8
Check. The fourth root of 16 is 2, and 23 = 8.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Watch out
Watch out: 8(2/3) is not 82 ÷ 3. Take the cube root of 8 first, which is 2, and then square it, to get 4.
Exam tip
Exam tip. Write the root and the power separately, as in the examples. This avoids mixing up the numerator and denominator.
Exercise 1B Fractional indices with a numerator
Non-calculatorMethod. Work out the root from the denominator, then the power from the numerator. The index (3/2) is the square root of 9, with the power 3.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.9(3/2) = (square root of 9)3
- Find the root. 3 multiplied by itself 2 times gives 9, so 3 is the square root.3 × 3 = 9, so the square root of 9 is 3
- Raise the root to the power. The numerator is 3, so raise 3 to the power 3.33 = 27
- Write the answer. 27
Check. The square root of 9 is 3, and 33 = 27.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (2/3) is the cube root of 8, with the power 2.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.8(2/3) = (cube root of 8)2
- Find the root. 2 multiplied by itself 3 times gives 8, so 2 is the cube root.2 × 2 × 2 = 8, so the cube root of 8 is 2
- Raise the root to the power. The numerator is 2, so raise 2 to the power 2.22 = 4
- Write the answer. 4
Check. The cube root of 8 is 2, and 22 = 4.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (3/4) is the fourth root of 16, with the power 3.
- Use the denominator as the root. The denominator 4 means the fourth root. The numerator is the power.16(3/4) = (fourth root of 16)3
- Find the root. 2 multiplied by itself 4 times gives 16, so 2 is the fourth root.2 × 2 × 2 × 2 = 16, so the fourth root of 16 is 2
- Raise the root to the power. The numerator is 3, so raise 2 to the power 3.23 = 8
- Write the answer. 8
Check. The fourth root of 16 is 2, and 23 = 8.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (3/2) is the square root of 25, with the power 3.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.25(3/2) = (square root of 25)3
- Find the root. 5 multiplied by itself 2 times gives 25, so 5 is the square root.5 × 5 = 25, so the square root of 25 is 5
- Raise the root to the power. The numerator is 3, so raise 5 to the power 3.53 = 125
- Write the answer. 125
Check. The square root of 25 is 5, and 53 = 125.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (2/3) is the cube root of 27, with the power 2.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.27(2/3) = (cube root of 27)2
- Find the root. 3 multiplied by itself 3 times gives 27, so 3 is the cube root.3 × 3 × 3 = 27, so the cube root of 27 is 3
- Raise the root to the power. The numerator is 2, so raise 3 to the power 2.32 = 9
- Write the answer. 9
Check. The cube root of 27 is 3, and 32 = 9.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (2/5) is the fifth root of 32, with the power 2.
- Use the denominator as the root. The denominator 5 means the fifth root. The numerator is the power.32(2/5) = (fifth root of 32)2
- Find the root. 2 multiplied by itself 5 times gives 32, so 2 is the fifth root.2 × 2 × 2 × 2 × 2 = 32, so the fifth root of 32 is 2
- Raise the root to the power. The numerator is 2, so raise 2 to the power 2.22 = 4
- Write the answer. 4
Check. The fifth root of 32 is 2, and 22 = 4.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (5/6) is the sixth root of 64, with the power 5.
- Use the denominator as the root. The denominator 6 means the sixth root. The numerator is the power.64(5/6) = (sixth root of 64)5
- Find the root. 2 multiplied by itself 6 times gives 64, so 2 is the sixth root.2 × 2 × 2 × 2 × 2 × 2 = 64, so the sixth root of 64 is 2
- Raise the root to the power. The numerator is 5, so raise 2 to the power 5.25 = 32
- Write the answer. 32
Check. The sixth root of 64 is 2, and 25 = 32.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (5/2) is the square root of 4, with the power 5.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.4(5/2) = (square root of 4)5
- Find the root. 2 multiplied by itself 2 times gives 4, so 2 is the square root.2 × 2 = 4, so the square root of 4 is 2
- Raise the root to the power. The numerator is 5, so raise 2 to the power 5.25 = 32
- Write the answer. 32
Check. The square root of 4 is 2, and 25 = 32.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (2/3) is the cube root of 1000, with the power 2.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.1000(2/3) = (cube root of 1000)2
- Find the root. 10 multiplied by itself 3 times gives 1000, so 10 is the cube root.10 × 10 × 10 = 1000, so the cube root of 1000 is 10
- Raise the root to the power. The numerator is 2, so raise 10 to the power 2.102 = 100
- Write the answer. 100
Check. The cube root of 1000 is 10, and 102 = 100.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (3/4) is the fourth root of 81, with the power 3.
- Use the denominator as the root. The denominator 4 means the fourth root. The numerator is the power.81(3/4) = (fourth root of 81)3
- Find the root. 3 multiplied by itself 4 times gives 81, so 3 is the fourth root.3 × 3 × 3 × 3 = 81, so the fourth root of 81 is 3
- Raise the root to the power. The numerator is 3, so raise 3 to the power 3.33 = 27
- Write the answer. 27
Check. The fourth root of 81 is 3, and 33 = 27.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (4/3) is the cube root of 8, with the power 4.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.8(4/3) = (cube root of 8)4
- Find the root. 2 multiplied by itself 3 times gives 8, so 2 is the cube root.2 × 2 × 2 = 8, so the cube root of 8 is 2
- Raise the root to the power. The numerator is 4, so raise 2 to the power 4.24 = 16
- Write the answer. 16
Check. The cube root of 8 is 2, and 24 = 16.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Take the cube root of 27 first, then square the result.
- Find the cube root of 27. The cube root of 27 is the number that multiplied three times gives 27.3 × 3 × 3 = 27, so the root is 3
- Raise the answer to the power 2. The numerator 2 is the power.32 = 9
- Write the result. 27(2/3) = 9
Check. If you square the root first, 27(2/3) = (272)(1/3) = 729(1/3) = 9 as well.
Watch out. Both orders give the same answer here, but the root first gives smaller numbers.
Method. Use the same method as before: the cube root first, then square it.
- Find the cube root of 27. The side length of the cube is the cube root of its volume.27(1/3) = 3 cm
- Square the side length. A face is a square, so its area is side × side.32 = 9
- Write the answer. 9 cm2
Check. A cube with side 3 cm has a face of 3 × 3 = 9 cm2.
Method. Work out the root from the denominator, then the power from the numerator. The index (3/2) is the square root of 25, with the power 3.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.25(3/2) = (square root of 25)3
- Find the root. 5 multiplied by itself 2 times gives 25, so 5 is the square root.5 × 5 = 25, so the square root of 25 is 5
- Raise the root to the power. The numerator is 3, so raise 5 to the power 3.53 = 125
- Write the answer. 125
Check. The square root of 25 is 5, and 53 = 125.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Negative indices
A negative index means one over the matching positive power. So 2(-3) = 1 ÷ 23 = 1/8.
A negative fractional index does both jobs. For 4(-1/2), work out the square root first: √4 = 2, then take the reciprocal: 4(-1/2) = 1/2.
Key ideas
- a(−n) = 1 ÷ an. A negative index gives a reciprocal.
- a(−m/n) = 1 ÷ (n-th root of a)m.
- A negative index does not make the answer negative.
- The index laws still work: 2(-3) × 25 = 22 = 4.
Method. A negative index gives a reciprocal: work out the positive power, then flip it to a fraction.
- Read the negative index. A negative index means one over the matching positive power.2−3 = 1 / 23
- Work out the positive power. 2 multiplied by itself 3 times.23 = 8
- Write the answer. The fraction has 1 on top.1/8
Check. 23 = 8, so the reciprocal is 1/8 = 1/8.
Watch out. A negative index does not make the answer negative. It makes it a small fraction.
Method. Work out the root from the denominator, then the power from the numerator. The index (−1/2) is the square root of 4, with the power 1 flipped to a fraction.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.4(−1/2) = 1 / ((square root of 4))1
- Find the root. 2 multiplied by itself 2 times gives 4, so 2 is the square root.2 × 2 = 4, so the square root of 4 is 2
- Take the reciprocal. A negative index means one over the positive power.1 / 2 = 1/2
- Write the answer. 1/2
Check. The square root of 4 is 2, and 21 = 2.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (−3/4) is the fourth root of 16, with the power 3 flipped to a fraction.
- Use the denominator as the root. The denominator 4 means the fourth root. The numerator is the power.16(−3/4) = 1 / ((fourth root of 16))3
- Find the root. 2 multiplied by itself 4 times gives 16, so 2 is the fourth root.2 × 2 × 2 × 2 = 16, so the fourth root of 16 is 2
- Take the reciprocal. A negative index means one over the positive power.1 / 8 = 1/8
- Write the answer. 1/8
Check. The fourth root of 16 is 2, and 23 = 8.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Watch out
Watch out: 2(-3) is 1/8, not −8. Also, 8(-1/3) is not −2. The negative index gives a reciprocal, and it is positive.
Exam tip
Exam tip. Deal with the sign of the index last: work out the positive power first, then take the reciprocal.
Exercise 1C Negative indices
Non-calculatorMethod. A negative index gives a reciprocal: work out the positive power, then flip it to a fraction.
- Read the negative index. A negative index means one over the matching positive power.2−3 = 1 / 23
- Work out the positive power. 2 multiplied by itself 3 times.23 = 8
- Write the answer. The fraction has 1 on top.1/8
Check. 23 = 8, so the reciprocal is 1/8 = 1/8.
Watch out. A negative index does not make the answer negative. It makes it a small fraction.
Method. A negative index gives a reciprocal: work out the positive power, then flip it to a fraction.
- Read the negative index. A negative index means one over the matching positive power.5−2 = 1 / 52
- Work out the positive power. 5 multiplied by itself 2 times.52 = 25
- Write the answer. The fraction has 1 on top.1/25
Check. 52 = 25, so the reciprocal is 1/25 = 1/25.
Watch out. A negative index does not make the answer negative. It makes it a small fraction.
Method. Work out the root from the denominator, then the power from the numerator. The index (−1/2) is the square root of 4, with the power 1 flipped to a fraction.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.4(−1/2) = 1 / ((square root of 4))1
- Find the root. 2 multiplied by itself 2 times gives 4, so 2 is the square root.2 × 2 = 4, so the square root of 4 is 2
- Take the reciprocal. A negative index means one over the positive power.1 / 2 = 1/2
- Write the answer. 1/2
Check. The square root of 4 is 2, and 21 = 2.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (−1/3) is the cube root of 8, with the power 1 flipped to a fraction.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.8(−1/3) = 1 / ((cube root of 8))1
- Find the root. 2 multiplied by itself 3 times gives 8, so 2 is the cube root.2 × 2 × 2 = 8, so the cube root of 8 is 2
- Take the reciprocal. A negative index means one over the positive power.1 / 2 = 1/2
- Write the answer. 1/2
Check. The cube root of 8 is 2, and 21 = 2.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (−3/4) is the fourth root of 16, with the power 3 flipped to a fraction.
- Use the denominator as the root. The denominator 4 means the fourth root. The numerator is the power.16(−3/4) = 1 / ((fourth root of 16))3
- Find the root. 2 multiplied by itself 4 times gives 16, so 2 is the fourth root.2 × 2 × 2 × 2 = 16, so the fourth root of 16 is 2
- Take the reciprocal. A negative index means one over the positive power.1 / 8 = 1/8
- Write the answer. 1/8
Check. The fourth root of 16 is 2, and 23 = 8.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (−1/2) is the square root of 81, with the power 1 flipped to a fraction.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.81(−1/2) = 1 / ((square root of 81))1
- Find the root. 9 multiplied by itself 2 times gives 81, so 9 is the square root.9 × 9 = 81, so the square root of 81 is 9
- Take the reciprocal. A negative index means one over the positive power.1 / 9 = 1/9
- Write the answer. 1/9
Check. The square root of 81 is 9, and 91 = 9.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (−3/2) is the square root of 9, with the power 3 flipped to a fraction.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.9(−3/2) = 1 / ((square root of 9))3
- Find the root. 3 multiplied by itself 2 times gives 9, so 3 is the square root.3 × 3 = 9, so the square root of 9 is 3
- Take the reciprocal. A negative index means one over the positive power.1 / 27 = 1/27
- Write the answer. 1/27
Check. The square root of 9 is 3, and 33 = 27.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (−2/5) is the fifth root of 32, with the power 2 flipped to a fraction.
- Use the denominator as the root. The denominator 5 means the fifth root. The numerator is the power.32(−2/5) = 1 / ((fifth root of 32))2
- Find the root. 2 multiplied by itself 5 times gives 32, so 2 is the fifth root.2 × 2 × 2 × 2 × 2 = 32, so the fifth root of 32 is 2
- Take the reciprocal. A negative index means one over the positive power.1 / 4 = 1/4
- Write the answer. 1/4
Check. The fifth root of 32 is 2, and 22 = 4.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (−2/3) is the cube root of 27, with the power 2 flipped to a fraction.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.27(−2/3) = 1 / ((cube root of 27))2
- Find the root. 3 multiplied by itself 3 times gives 27, so 3 is the cube root.3 × 3 × 3 = 27, so the cube root of 27 is 3
- Take the reciprocal. A negative index means one over the positive power.1 / 9 = 1/9
- Write the answer. 1/9
Check. The cube root of 27 is 3, and 32 = 9.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Use the index law for multiplying powers with the same base: add the indices.
- Add the indices. Both powers have base 2, so add the indices: −3 + 5 = 2.2(-3) × 25 = 2(-3 + 5) = 22
- Work out 22. Two 2s multiplied together.2 × 2 = 4
- Write the answer. 4
Check. 2(-3) = 1/8, and 25 = 32. Then 32/8 = 4.
Method. Work out each term, then add them.
- Work out 4(-1/2). The square root of 4 is 2. The negative index gives the reciprocal.4(-1/2) = 1/2
- Work out 8(1/3). The cube root of 8 is 2.8(1/3) = 2
- Add the two values. Add the fraction and the whole number.1/2 + 2 = 2 1/2
- Write as a fraction. Two and a half is the same as 5/2.5/2
Check. 5/2 = 2.5, and 0.5 + 2 = 2.5.
Method. Work out both values as fractions and compare them.
- Work out 3(-1). One over 3 to the power 1.3(-1) = 1/3
- Work out 3(-2). One over 3 squared, which is 9.3(-2) = 1/9
- Compare. The bigger the denominator, the smaller the fraction.1/3 > 1/9
- Write the answer. 3(-1) is bigger
Check. 1/3 is about 0.33 and 1/9 is about 0.11.
Watch out. A bigger negative index makes a smaller number, not a bigger one.
Method. Work out the root from the denominator, then the power from the numerator. The index (−3/2) is the square root of 9, with the power 3 flipped to a fraction.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.9(−3/2) = 1 / ((square root of 9))3
- Find the root. 3 multiplied by itself 2 times gives 9, so 3 is the square root.3 × 3 = 9, so the square root of 9 is 3
- Take the reciprocal. A negative index means one over the positive power.1 / 27 = 1/27
- Write the answer. 1/27
Check. The square root of 9 is 3, and 33 = 27.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Multiply the answer to part (a) by 27.
- Use the answer from part (a). The value of 9(-3/2) is 1/27.1/27 × 27
- Cancel the 27. The 27 on the top and bottom cancel.= 1
Check. 27 ÷ 27 = 1.
Simplifying and adding surds
A surd is a root that cannot be written exactly as a whole number, such as √2 or √50. To simplify a surd, find a square number that divides the number under the root. Then split the root: √50 = √(25 × 2) = √25 × √2 = 5√2.
Like surds have the same number under the root, like 3√2 and 5√2. Only like surds can be added or subtracted: 3√2 + 5√2 = 8√2.
Key ideas
- √(a × b) = √a × √b. Use it to split off a square number.
- Simplify each surd first, then add or subtract the like surds.
- Like surds have the same number under the root.
- You cannot add surds with different numbers under the root: √2 + √3 cannot be simplified.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√72 = √(62 × 2) = 6√2
- Collect like surds. Only surds with the same number under the root can be added or subtracted.6√2
- Write the answer. 6√2
Check. Both forms are about 8.485.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.2√12 = 4√3
√27 = √(32 × 3) = 3√3 - Collect like surds. Only surds with the same number under the root can be added or subtracted.4√3 + 3√3 = 7√3
- Write the answer. 7√3
Check. Both forms are about 12.124.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√80 = √(42 × 5) = 4√5
−√20 = −2√5
√5 is already in its simplest form - Collect like surds. Only surds with the same number under the root can be added or subtracted.4√5 − 2√5 + √5 = 3√5
- Write the answer. 3√5
Check. Both forms are about 6.708.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Watch out
Watch out: √2 + √2 = 2√2, not 2√4. You can only add surds that are the same. Also √8 is not 8√2; √8 = 2√2.
Exam tip
Exam tip. Look for the largest square number that divides the number under the root, such as 4, 9, 16, 25 or 36.
Exercise 1D Simplifying and adding surds
Non-calculatorMethod. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√200 = √(102 × 2) = 10√2
- Collect like surds. Only surds with the same number under the root can be added or subtracted.10√2
- Write the answer. 10√2
Check. Both forms are about 14.142.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√108 = √(62 × 3) = 6√3
- Collect like surds. Only surds with the same number under the root can be added or subtracted.6√3
- Write the answer. 6√3
Check. Both forms are about 10.392.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√147 = √(72 × 3) = 7√3
- Collect like surds. Only surds with the same number under the root can be added or subtracted.7√3
- Write the answer. 7√3
Check. Both forms are about 12.124.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√180 = √(62 × 5) = 6√5
- Collect like surds. Only surds with the same number under the root can be added or subtracted.6√5
- Write the answer. 6√5
Check. Both forms are about 13.416.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√320 = √(82 × 5) = 8√5
- Collect like surds. Only surds with the same number under the root can be added or subtracted.8√5
- Write the answer. 8√5
Check. Both forms are about 17.889.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√98 = √(72 × 2) = 7√2
- Collect like surds. Only surds with the same number under the root can be added or subtracted.7√2
- Write the answer. 7√2
Check. Both forms are about 9.899.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√75 = √(52 × 3) = 5√3
- Collect like surds. Only surds with the same number under the root can be added or subtracted.5√3
- Write the answer. 5√3
Check. Both forms are about 8.660.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√72 = √(62 × 2) = 6√2
- Collect like surds. Only surds with the same number under the root can be added or subtracted.6√2
- Write the answer. 6√2
Check. Both forms are about 8.485.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.3√2 is already in its simplest form
5√2 is already in its simplest form - Collect like surds. Only surds with the same number under the root can be added or subtracted.3√2 + 5√2 = 8√2
- Write the answer. 8√2
Check. Both forms are about 11.314.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√12 = √(22 × 3) = 2√3
√27 = √(32 × 3) = 3√3
−√48 = −4√3 - Collect like surds. Only surds with the same number under the root can be added or subtracted.2√3 + 3√3 − 4√3 = √3
- Write the answer. √3
Check. Both forms are about 1.732.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.4√50 = 20√2
−3√8 = −6√2 - Collect like surds. Only surds with the same number under the root can be added or subtracted.20√2 − 6√2 = 14√2
- Write the answer. 14√2
Check. Both forms are about 19.799.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√18 = √(32 × 2) = 3√2
√8 = √(22 × 2) = 2√2 - Collect like surds. Only surds with the same number under the root can be added or subtracted.3√2 + 2√2 = 5√2
- Write the answer. 5√2
Check. Both forms are about 7.071.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√72 = √(62 × 2) = 6√2
√50 = √(52 × 2) = 5√2 - Collect like surds. Only surds with the same number under the root can be added or subtracted.6√2 + 5√2 = 11√2
- Write the answer. 11√2
Check. Both forms are about 15.556.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Multiplying and expanding brackets with surds
Expanding brackets with surds works in the same way as expanding brackets with letters. Multiply every term in the first bracket by every term in the second. Then simplify each product, and collect the like terms.
Remember that √a × √a = a. For example, √3 × √3 = 3, and √3 × √12 = √36 = 6.
Key ideas
- Expand brackets with surds in the same way as brackets with letters: multiply every term by every term.
- √a × √a = a, and √a × √b = √(ab).
- Simplify each surd, then collect the like terms.
- The difference of two squares: (p + √q)(p − √q) = p2 − q, which has no surd.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.2 × (1) = 2
2 × (−√3) = −2√3
√3 × (1) = √3
√3 × (−√3) = −√9 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.−√9 = −3
- Collect like terms. Add the whole numbers, and add the surds that match.2 − 2√3 + √3 − 3 = −1 − √3
- Write the answer. −1 − √3
Check. Both sides are about -2.732.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.√5 × (√5) = √25
√5 × (−2) = −2√5
2 × (√5) = 2√5
2 × (−2) = −4 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√25 = 5
- Collect like terms. Add the whole numbers, and add the surds that match.5 − 2√5 + 2√5 − 4 = 1
- Write the answer. 1
Check. Both sides are about 1.000.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.3 × (3) = 9
3 × (√2) = 3√2
√2 × (3) = 3√2
√2 × (√2) = √4 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√4 = 2
- Collect like terms. Add the whole numbers, and add the surds that match.9 + 3√2 + 3√2 + 2 = 11 + 6√2
- Write the answer. 11 + 6√2
Check. Both sides are about 19.485.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Watch out
Watch out: (3 + √2)2 is not 9 + 2 = 11. The middle terms 3√2 + 3√2 = 6√2 are missing from that idea.
Exam tip
Exam tip. Write out all four products before you simplify. Missing a term is the most common error.
Exercise 1E Expanding brackets with surds
Non-calculatorMethod. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.√2 × (√2) = √4
√2 × (−1) = −√2
1 × (√2) = √2
1 × (−1) = −1 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√4 = 2
- Collect like terms. Add the whole numbers, and add the surds that match.2 − √2 + √2 − 1 = 1
- Write the answer. 1
Check. Both sides are about 1.000.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.3 × (1) = 3
3 × (√2) = 3√2
√2 × (1) = √2
√2 × (√2) = √4 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√4 = 2
- Collect like terms. Add the whole numbers, and add the surds that match.3 + 3√2 + √2 + 2 = 5 + 4√2
- Write the answer. 5 + 4√2
Check. Both sides are about 10.657.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.2 × (2) = 4
2 × (−√3) = −2√3
√3 × (2) = 2√3
√3 × (−√3) = −√9 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.−√9 = −3
- Collect like terms. Add the whole numbers, and add the surds that match.4 − 2√3 + 2√3 − 3 = 1
- Write the answer. 1
Check. Both sides are about 1.000.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.√7 × (√7) = √49
√7 × (2) = 2√7
1 × (√7) = √7
1 × (2) = 2 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√49 = 7
- Collect like terms. Add the whole numbers, and add the surds that match.7 + 2√7 + √7 + 2 = 9 + 3√7
- Write the answer. 9 + 3√7
Check. Both sides are about 16.937.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.3√2 × (√2) = 3√4
3√2 × (√8) = 3√16 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.3√4 = 6
3√16 = 12 - Collect like terms. Add the whole numbers, and add the surds that match.6 + 12 = 18
- Write the answer. 18
Check. Both sides are about 18.000.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.√5 × (√5) = √25
√5 × (3) = 3√5
−1 × (√5) = −√5
−1 × (3) = −3 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√25 = 5
- Collect like terms. Add the whole numbers, and add the surds that match.5 + 3√5 − √5 − 3 = 2 + 2√5
- Write the answer. 2 + 2√5
Check. Both sides are about 6.472.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.4 × (1) = 4
4 × (−√6) = −4√6
√6 × (1) = √6
√6 × (−√6) = −√36 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.−√36 = −6
- Collect like terms. Add the whole numbers, and add the surds that match.4 − 4√6 + √6 − 6 = −2 − 3√6
- Write the answer. −2 − 3√6
Check. Both sides are about -9.348.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.√3 × (√3) = √9
√3 × (2) = 2√3
2 × (√3) = 2√3
2 × (2) = 4 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√9 = 3
- Collect like terms. Add the whole numbers, and add the surds that match.3 + 2√3 + 2√3 + 4 = 7 + 4√3
- Write the answer. 7 + 4√3
Check. Both sides are about 13.928.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.2√2 × (3√2) = 6√4
2√2 × (1) = 2√2 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.6√4 = 12
- Collect like terms. Add the whole numbers, and add the surds that match.12 + 2√2 = 12 + 2√2
- Write the answer. 12 + 2√2
Check. Both sides are about 14.828.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.1 × (1) = 1
1 × (−2√2) = −2√2
2√2 × (1) = 2√2
2√2 × (−2√2) = −4√4 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.−4√4 = −8
- Collect like terms. Add the whole numbers, and add the surds that match.1 − 2√2 + 2√2 − 8 = −7
- Write the answer. −7
Check. Both sides are about -7.000.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.√10 × (√10) = √100
√10 × (−1) = −√10
1 × (√10) = √10
1 × (−1) = −1 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√100 = 10
- Collect like terms. Add the whole numbers, and add the surds that match.10 − √10 + √10 − 1 = 9
- Write the answer. 9
Check. Both sides are about 9.000.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.√3 × (√3) = √9
√3 × (1) = √3
1 × (√3) = √3
1 × (1) = 1 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√9 = 3
- Collect like terms. Add the whole numbers, and add the surds that match.3 + √3 + √3 + 1 = 4 + 2√3
- Write the answer. 4 + 2√3
Check. Both sides are about 7.464.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Extension: rationalising denominators
Rationalising a denominator means writing the fraction so that the bottom has no surd. For a single surd on the bottom, multiply the top and bottom by that surd.
If the bottom is a surd with a whole number, such as 2 + √3, multiply top and bottom by the conjugate, 2 − √3. The bottom becomes a whole number, because (2 + √3)(2 − √3) = 4 − 3 = 1.
Key ideas
- Extension: this topic is more advanced than most KS3 work. Check with your teacher.
- For k/√n, multiply the top and bottom by √n.
- For k/(a + √b), multiply the top and bottom by the conjugate (a − √b).
- The conjugate changes only the sign between the two terms.
Method. Multiply top and bottom by the surd on the bottom, so the bottom has no surd.
- Multiply the top and bottom by √3. Multiplying by √3 / √3 changes the form but not the value. The bottom becomes a whole number.1/√3 = 1√3/(√3 × √3) = 1√3/3
- Simplify the fraction. Divide the top and the bottom by any common factor.1√3/3, which cannot be simplified further
- Write the answer. √3/3
Check. √3/3 is about 0.577, and 1/√3 is about 0.577.
Watch out. Do not multiply only the bottom. Whatever you do to the bottom you must also do to the top.
Method. Multiply top and bottom by the surd on the bottom, so the bottom has no surd.
- Multiply the top and bottom by √2. Multiplying by √2 / √2 changes the form but not the value. The bottom becomes a whole number.6/√2 = 6√2/(√2 × √2) = 6√2/2
- Simplify the fraction. Divide the top and the bottom by any common factor.6√2/2 = 3√2
- Write the answer. 3√2
Check. 3√2 is about 4.243, and 6/√2 is about 4.243.
Watch out. Do not multiply only the bottom. Whatever you do to the bottom you must also do to the top.
Method. Multiply by the conjugate to remove the surd from the bottom, then simplify.
- Multiply the top and bottom by the conjugate. The conjugate of 2 + √3 is 2 − √3. Multiplying the bottom by it removes the surd.1/(2 + √3) = 1(2 − √3) / ((2 + √3)(2 − √3))
- Expand the top. Multiply 1 by each term in the bracket.1(2 − √3) = 2 − √3
- Expand the bottom. Difference of two squares: (p + √q)(p − √q) = p² − q.2² − 3 = 1
- Simplify the fraction. Divide every term on the top by the same number as the bottom, and tidy the sign.(2 − √3)/(1) = 2 − √3
- Write the answer. 2 − √3
Check. 1/(2 + √3) is about 0.268, and the answer is about 0.268.
Watch out. The conjugate changes only the sign between the two terms. Use the difference of two squares on the bottom.
Watch out
Watch out: changing the bottom alone changes the value. Whatever you multiply the bottom by, you must multiply the top by the same number.
Exam tip
Exam tip. Check your answer by working out the decimal value of both the fraction and your answer. They should match.
Exercise 1F Rationalising denominators (extension)
Non-calculatorMethod. Multiply top and bottom by the surd on the bottom, so the bottom has no surd.
- Multiply the top and bottom by √3. Multiplying by √3 / √3 changes the form but not the value. The bottom becomes a whole number.1/√3 = 1√3/(√3 × √3) = 1√3/3
- Simplify the fraction. Divide the top and the bottom by any common factor.1√3/3, which cannot be simplified further
- Write the answer. √3/3
Check. √3/3 is about 0.577, and 1/√3 is about 0.577.
Watch out. Do not multiply only the bottom. Whatever you do to the bottom you must also do to the top.
Method. Multiply top and bottom by the surd on the bottom, so the bottom has no surd.
- Multiply the top and bottom by √5. Multiplying by √5 / √5 changes the form but not the value. The bottom becomes a whole number.10/√5 = 10√5/(√5 × √5) = 10√5/5
- Simplify the fraction. Divide the top and the bottom by any common factor.10√5/5 = 2√5
- Write the answer. 2√5
Check. 2√5 is about 4.472, and 10/√5 is about 4.472.
Watch out. Do not multiply only the bottom. Whatever you do to the bottom you must also do to the top.
Method. Multiply top and bottom by the surd on the bottom, so the bottom has no surd.
- Multiply the top and bottom by √2. Multiplying by √2 / √2 changes the form but not the value. The bottom becomes a whole number.6/√2 = 6√2/(√2 × √2) = 6√2/2
- Simplify the fraction. Divide the top and the bottom by any common factor.6√2/2 = 3√2
- Write the answer. 3√2
Check. 3√2 is about 4.243, and 6/√2 is about 4.243.
Watch out. Do not multiply only the bottom. Whatever you do to the bottom you must also do to the top.
Method. Multiply by the conjugate to remove the surd from the bottom, then simplify.
- Multiply the top and bottom by the conjugate. The conjugate of 2 + √3 is 2 − √3. Multiplying the bottom by it removes the surd.1/(2 + √3) = 1(2 − √3) / ((2 + √3)(2 − √3))
- Expand the top. Multiply 1 by each term in the bracket.1(2 − √3) = 2 − √3
- Expand the bottom. Difference of two squares: (p + √q)(p − √q) = p² − q.2² − 3 = 1
- Simplify the fraction. Divide every term on the top by the same number as the bottom, and tidy the sign.(2 − √3)/(1) = 2 − √3
- Write the answer. 2 − √3
Check. 1/(2 + √3) is about 0.268, and the answer is about 0.268.
Watch out. The conjugate changes only the sign between the two terms. Use the difference of two squares on the bottom.
Method. Multiply by the conjugate to remove the surd from the bottom, then simplify.
- Multiply the top and bottom by the conjugate. The conjugate of 1 + √5 is 1 − √5. Multiplying the bottom by it removes the surd.2/(1 + √5) = 2(1 − √5) / ((1 + √5)(1 − √5))
- Expand the top. Multiply 2 by each term in the bracket.2(1 − √5) = 2 − 2√5
- Expand the bottom. Difference of two squares: (p + √q)(p − √q) = p² − q.1² − 5 = −4
- Simplify the fraction. Divide every term on the top by the same number as the bottom, and tidy the sign.(2 − 2√5)/(−4) = (−1 + √5)/2
- Write the answer. (−1 + √5)/2
Check. 2/(1 + √5) is about 0.618, and the answer is about 0.618.
Watch out. The conjugate changes only the sign between the two terms. Use the difference of two squares on the bottom.
Method. Multiply by the conjugate to remove the surd from the bottom, then simplify.
- Multiply the top and bottom by the conjugate. The conjugate of 3 + √2 is 3 − √2. Multiplying the bottom by it removes the surd.5/(3 + √2) = 5(3 − √2) / ((3 + √2)(3 − √2))
- Expand the top. Multiply 5 by each term in the bracket.5(3 − √2) = 15 − 5√2
- Expand the bottom. Difference of two squares: (p + √q)(p − √q) = p² − q.3² − 2 = 7
- Simplify the fraction. Divide every term on the top by the same number as the bottom, and tidy the sign.(15 − 5√2)/(7) = (15 − 5√2)/7
- Write the answer. (15 − 5√2)/7
Check. 5/(3 + √2) is about 1.133, and the answer is about 1.133.
Watch out. The conjugate changes only the sign between the two terms. Use the difference of two squares on the bottom.
Chapter summary
| Skill | Method |
|---|---|
| Fractional index a(1/n) | The n-th root of a. For example 27(1/3) = 3. |
| Index a(m/n) | Take the n-th root first, then raise to the power m. |
| Negative index a(−n) | One over an. The answer is positive. |
| Simplify a surd | Split off the largest square factor: √50 = 5√2. |
| Add surds | Simplify first, then add only the like surds. |
| Expand with surds | Multiply every term by every term, simplify each surd, then collect like terms. |
| Extension: rationalise | Multiply top and bottom by √n, or by the conjugate a − √b. |
Can you…?
- I can work out a fractional index as a root and a power
- I can work out a negative index, including a negative fractional index
- I can simplify a surd and add or subtract like surds
- I can expand two brackets that contain surds and simplify the answer
- Extension: I can rationalise a denominator with a single surd or a conjugate pair
Key terms
- index (power)
- the small number that shows how many times the base is multiplied by itself
- surd
- a root that cannot be written exactly as a whole number
- like surds
- surds with the same number under the root
- conjugate
- for a + √b, the conjugate is a − √b
- rationalise
- rewrite a fraction so the denominator has no surd
Exercise 1Z Check-up
Non-calculatorMethod. Work out the root from the denominator, then the power from the numerator. The index (1/2) is the square root of 25, with the power 1.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.25(1/2) = (square root of 25)1
- Find the root. 5 multiplied by itself 2 times gives 25, so 5 is the square root.5 × 5 = 25, so the square root of 25 is 5
- Raise the root to the power. The numerator is 1, so raise 5 to the power 1.51 = 5
- Write the answer. 5
Check. The square root of 25 is 5, and 51 = 5.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (2/3) is the cube root of 125, with the power 2.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.125(2/3) = (cube root of 125)2
- Find the root. 5 multiplied by itself 3 times gives 125, so 5 is the cube root.5 × 5 × 5 = 125, so the cube root of 125 is 5
- Raise the root to the power. The numerator is 2, so raise 5 to the power 2.52 = 25
- Write the answer. 25
Check. The cube root of 125 is 5, and 52 = 25.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (−1/4) is the fourth root of 16, with the power 1 flipped to a fraction.
- Use the denominator as the root. The denominator 4 means the fourth root. The numerator is the power.16(−1/4) = 1 / ((fourth root of 16))1
- Find the root. 2 multiplied by itself 4 times gives 16, so 2 is the fourth root.2 × 2 × 2 × 2 = 16, so the fourth root of 16 is 2
- Take the reciprocal. A negative index means one over the positive power.1 / 2 = 1/2
- Write the answer. 1/2
Check. The fourth root of 16 is 2, and 21 = 2.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. A negative index gives a reciprocal: work out the positive power, then flip it to a fraction.
- Read the negative index. A negative index means one over the matching positive power.2−4 = 1 / 24
- Work out the positive power. 2 multiplied by itself 4 times.24 = 16
- Write the answer. The fraction has 1 on top.1/16
Check. 24 = 16, so the reciprocal is 1/16 = 1/16.
Watch out. A negative index does not make the answer negative. It makes it a small fraction.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√80 = √(42 × 5) = 4√5
- Collect like surds. Only surds with the same number under the root can be added or subtracted.4√5
- Write the answer. 4√5
Check. Both forms are about 8.944.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.2√18 = 6√2
√32 = √(42 × 2) = 4√2 - Collect like surds. Only surds with the same number under the root can be added or subtracted.6√2 + 4√2 = 10√2
- Write the answer. 10√2
Check. Both forms are about 14.142.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.√2 × (√8) = √16
- Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√16 = 4
- Collect like terms. Add the whole numbers, and add the surds that match.4 = 4
- Write the answer. 4
Check. Both sides are about 4.000.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.1 × (1) = 1
1 × (−2√2) = −2√2
2√2 × (1) = 2√2
2√2 × (−2√2) = −4√4 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.−4√4 = −8
- Collect like terms. Add the whole numbers, and add the surds that match.1 − 2√2 + 2√2 − 8 = −7
- Write the answer. −7
Check. Both sides are about -7.000.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply top and bottom by the surd on the bottom, so the bottom has no surd.
- Multiply the top and bottom by √2. Multiplying by √2 / √2 changes the form but not the value. The bottom becomes a whole number.4/√2 = 4√2/(√2 × √2) = 4√2/2
- Simplify the fraction. Divide the top and the bottom by any common factor.4√2/2 = 2√2
- Write the answer. 2√2
Check. 2√2 is about 2.828, and 4/√2 is about 2.828.
Watch out. Do not multiply only the bottom. Whatever you do to the bottom you must also do to the top.
Method. Multiply by the conjugate to remove the surd from the bottom, then simplify.
- Multiply the top and bottom by the conjugate. The conjugate of 4 + √5 is 4 − √5. Multiplying the bottom by it removes the surd.1/(4 + √5) = 1(4 − √5) / ((4 + √5)(4 − √5))
- Expand the top. Multiply 1 by each term in the bracket.1(4 − √5) = 4 − √5
- Expand the bottom. Difference of two squares: (p + √q)(p − √q) = p² − q.4² − 5 = 11
- Simplify the fraction. Divide every term on the top by the same number as the bottom, and tidy the sign.(4 − √5)/(11) = (4 − √5)/11
- Write the answer. (4 − √5)/11
Check. 1/(4 + √5) is about 0.160, and the answer is about 0.160.
Watch out. The conjugate changes only the sign between the two terms. Use the difference of two squares on the bottom.
Total 21 marks. Allow about 21 minutes.
Exercise 1X Practice test questions
Non-calculatorMethod. Work out the root from the denominator, then the power from the numerator. The index (1/2) is the square root of 49, with the power 1.
- Use the denominator as the root. The denominator 2 means the square root. The numerator is the power.49(1/2) = (square root of 49)1
- Find the root. 7 multiplied by itself 2 times gives 49, so 7 is the square root.7 × 7 = 49, so the square root of 49 is 7
- Raise the root to the power. The numerator is 1, so raise 7 to the power 1.71 = 7
- Write the answer. 7
Check. The square root of 49 is 7, and 71 = 7.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (1/3) is the cube root of 216, with the power 1.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.216(1/3) = (cube root of 216)1
- Find the root. 6 multiplied by itself 3 times gives 216, so 6 is the cube root.6 × 6 × 6 = 216, so the cube root of 216 is 6
- Raise the root to the power. The numerator is 1, so raise 6 to the power 1.61 = 6
- Write the answer. 6
Check. The cube root of 216 is 6, and 61 = 6.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (3/5) is the fifth root of 32, with the power 3.
- Use the denominator as the root. The denominator 5 means the fifth root. The numerator is the power.32(3/5) = (fifth root of 32)3
- Find the root. 2 multiplied by itself 5 times gives 32, so 2 is the fifth root.2 × 2 × 2 × 2 × 2 = 32, so the fifth root of 32 is 2
- Raise the root to the power. The numerator is 3, so raise 2 to the power 3.23 = 8
- Write the answer. 8
Check. The fifth root of 32 is 2, and 23 = 8.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Work out the root from the denominator, then the power from the numerator. The index (−2/3) is the cube root of 27, with the power 2 flipped to a fraction.
- Use the denominator as the root. The denominator 3 means the cube root. The numerator is the power.27(−2/3) = 1 / ((cube root of 27))2
- Find the root. 3 multiplied by itself 3 times gives 27, so 3 is the cube root.3 × 3 × 3 = 27, so the cube root of 27 is 3
- Take the reciprocal. A negative index means one over the positive power.1 / 9 = 1/9
- Write the answer. 1/9
Check. The cube root of 27 is 3, and 32 = 9.
Watch out. Do the root first, then the power. Writing the power first gives a number that is too big to work with.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.√75 = √(52 × 3) = 5√3
√12 = √(22 × 3) = 2√3 - Collect like surds. Only surds with the same number under the root can be added or subtracted.5√3 + 2√3 = 7√3
- Write the answer. 7√3
Check. Both forms are about 12.124.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Simplify each surd first, then add or subtract the surds that match.
- Simplify each surd. Find a square number that divides the number under the root, then take its root out. The root of a product is the product of the roots.4√45 = 12√5
−2√20 = −4√5 - Collect like surds. Only surds with the same number under the root can be added or subtracted.12√5 − 4√5 = 8√5
- Write the answer. 8√5
Check. Both forms are about 17.889.
Watch out. You cannot add surds with different numbers under the root. √2 + √3 does not equal √5.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.√2 × (√2) = √4
√2 × (−1) = −√2
5 × (√2) = 5√2
5 × (−1) = −5 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√4 = 2
- Collect like terms. Add the whole numbers, and add the surds that match.2 − √2 + 5√2 − 5 = −3 + 4√2
- Write the answer. −3 + 4√2
Check. Both sides are about 2.657.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.√7 × (√7) = √49
√7 × (2) = 2√7
−2 × (√7) = −2√7
−2 × (2) = −4 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√49 = 7
- Collect like terms. Add the whole numbers, and add the surds that match.7 + 2√7 − 2√7 − 4 = 3
- Write the answer. 3
Check. Both sides are about 3.000.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.
Method. The side length is the square root of the area. Simplify the surd.
- Write the side length. The area of a square is side × side, so the side is the square root of 50.side = √50 cm
- Simplify the surd. 50 = 25 × 2, and 25 is a square number.√50 = √25 × √2 = 5√2
- Write the answer. 5√2 cm
Check. 5√2 is about 7.07, and 7.07 × 7.07 is about 50.
Method. Multiply out the brackets term by term, then simplify and collect like terms.
- Multiply every term in the first bracket by every term in the second. Each term in one bracket multiplies each term in the other, as with double brackets.√2 × (√2) = √4
√2 × (1) = √2
1 × (√2) = √2
1 × (1) = 1 - Simplify the surds. Use √a × √b = √(ab), then take out square factors where you can.√4 = 2
- Collect like terms. Add the whole numbers, and add the surds that match.2 + √2 + √2 + 1 = 3 + 2√2
- Write the answer. 3 + 2√2
Check. Both sides are about 5.828.
Watch out. Remember the minus sign: −2 × √3 is −2√3, and √3 × √3 = 3.