GCSE Maths: Rationalising the denominator and exact answers (Higher)
Part of the GCSE Maths standard form, indices and surds chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Rationalising the denominator and exact answers
A fraction with a surd in its denominator, such as 1/√2, is awkward to work with, and it is not considered fully simplified. Rationalising the denominator means rewriting the fraction so that the bottom is a whole number. We use the fact that multiplying the top and the bottom of a fraction by the same number does not change its value, and that √2 × √2 = 2:
1√2 = 1√2 × √2√2 = √22
Type 1: a single surd on the bottom (such as 6/√3 or 3/(2√5)). Multiply the top and bottom by that surd, then simplify: 6/√3 = 6√3/3 = 2√3.
Type 2: a whole number plus or minus a surd (such as 1/(3 + √5)). Multiplying by the surd alone does not work, because the bottom has two terms. Instead multiply top and bottom by the conjugate: the same two terms with the sign in the middle changed. The conjugate of 3 + √5 is 3 − √5. By the difference of two squares, (3 + √5)(3 − √5) = 9 − 5 = 4, a whole number. So 1/(3 + √5) = (3 − √5)/4.
Surds also give exact answers in geometry. In Pythagoras, if c2 = 52 then c = √52 = 2√13 cm exactly. Give exact answers unless the question asks you to round.
Key ideas
- Rationalise: multiply top and bottom by the same thing so the bottom becomes a whole number.
- For k/(c√n) multiply top and bottom by √n.
- For k/(a + √b) multiply by the conjugate (a − √b); the bottom becomes a2 − b.
- Always simplify the final fraction (divide the top and bottom by common factors).
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.6√3
- Multiply the top and the bottom by √3. √3 ÷ √3 = 1, so multiplying by it does not change the value of the fraction. We choose √3 because √3 × √3 = 3, a whole number.6√3 × √3√3 = 6√3√3×√3
- Work out the top and the bottom. Top: multiply each term by √3. Bottom: √3 × √3 = 3.6√33
- Simplify the fraction. Divide the top and the bottom by any number that goes into every term.2√3
Check. Decimals: 3.464 before and 3.464 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.32√5
- Multiply the top and the bottom by √5. √5 ÷ √5 = 1, so multiplying by it does not change the value of the fraction. We choose √5 because √5 × √5 = 5, a whole number.32√5 × √5√5 = 3√52×√5×√5
- Work out the top and the bottom. Top: multiply each term by √5. Bottom: √5 × √5 = 5.3√510
- The fraction cannot be simplified further. 3√510
Check. Decimals: 0.671 before and 0.671 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.1 + √3√2
- Multiply the top and the bottom by √2. √2 ÷ √2 = 1, so multiplying by it does not change the value of the fraction. We choose √2 because √2 × √2 = 2, a whole number.1 + √3√2 × √2√2 = (1 + √3)√2√2×√2
- Work out the top and the bottom. Top: multiply each term by √2. Bottom: √2 × √2 = 2.√2 + √62
Top: (1 + √3)√2 = √2 + √6 - The fraction cannot be simplified further. √2 + √62
Check. Decimals: 1.932 before and 1.932 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.13 + √5
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 3 + √5 conjugate: 3 − √5
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.1(3 + √5) × (3 − √5)(3 − √5)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(3 + √5)(3 − √5) = 32 − (√5)2 = 9 − 5 = 4
- Expand the top. Multiply the top by the conjugate, term by term.1 × (3 − √5) = 3 − √5
- Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.3 − √54 = 3 − √54
Check. Decimals: 0.191 before and 0.191 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.64 − √2
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 4 − √2 conjugate: 4 + √2
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.6(4 − √2) × (4 + √2)(4 + √2)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(4 − √2)(4 + √2) = 42 − (√2)2 = 16 − 2 = 14
- Expand the top. Multiply the top by the conjugate, term by term.6 × (4 + √2) = 24 + 6√2
- Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.24 + 6√214 = 12 + 3√27
Check. Decimals: 2.320 before and 2.320 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Use Pythagoras to get the square of the length, then take the square root and leave it as a simplified surd.
- Write Pythagoras' theorem. In a right-angled triangle the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides.c2 = a2 + b2
- Substitute the two shorter sides. c2 = 42 + 62 = 16 + 36 = 52
- Take the square root. Leave it as a surd, so that the answer is exact.side = √52
- Simplify the surd. The largest square factor of 52 is 4.√52 = √4 × 13 = 2√13
- Write the answer with units. This is the exact length.2√13 cm
Check. Square the answer: (2√13)2 = 52. As a decimal the length is about 7.21 cm. ✓
Watch out. Do not round if the question asks for an exact answer. Simplify the surd.
Watch out
Multiply the top and the bottom. For a bottom like 3 + √5, use the conjugate 3 − √5, not just √5. Check that the surd really disappears from the bottom, and simplify the fraction at the end.Exercise 6Q Rationalising the denominator
Non-calculatorKey words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.6√3
- Multiply the top and the bottom by √3. √3 ÷ √3 = 1, so multiplying by it does not change the value of the fraction. We choose √3 because √3 × √3 = 3, a whole number.6√3 × √3√3 = 6√3√3×√3
- Work out the top and the bottom. Top: multiply each term by √3. Bottom: √3 × √3 = 3.6√33
- Simplify the fraction. Divide the top and the bottom by any number that goes into every term.2√3
Check. Decimals: 3.464 before and 3.464 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.1√7
- Multiply the top and the bottom by √7. √7 ÷ √7 = 1, so multiplying by it does not change the value of the fraction. We choose √7 because √7 × √7 = 7, a whole number.1√7 × √7√7 = 1√7√7×√7
- Work out the top and the bottom. Top: multiply each term by √7. Bottom: √7 × √7 = 7.√77
- The fraction cannot be simplified further. √77
Check. Decimals: 0.378 before and 0.378 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.10√5
- Multiply the top and the bottom by √5. √5 ÷ √5 = 1, so multiplying by it does not change the value of the fraction. We choose √5 because √5 × √5 = 5, a whole number.10√5 × √5√5 = 10√5√5×√5
- Work out the top and the bottom. Top: multiply each term by √5. Bottom: √5 × √5 = 5.10√55
- Simplify the fraction. Divide the top and the bottom by any number that goes into every term.2√5
Check. Decimals: 4.472 before and 4.472 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.42√3
- Multiply the top and the bottom by √3. √3 ÷ √3 = 1, so multiplying by it does not change the value of the fraction. We choose √3 because √3 × √3 = 3, a whole number.42√3 × √3√3 = 4√32×√3×√3
- Work out the top and the bottom. Top: multiply each term by √3. Bottom: √3 × √3 = 3.4√36
- Simplify the fraction. Divide the top and the bottom by any number that goes into every term.2√33
Check. Decimals: 1.155 before and 1.155 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.12√6
- Multiply the top and the bottom by √6. √6 ÷ √6 = 1, so multiplying by it does not change the value of the fraction. We choose √6 because √6 × √6 = 6, a whole number.12√6 × √6√6 = 12√6√6×√6
- Work out the top and the bottom. Top: multiply each term by √6. Bottom: √6 × √6 = 6.12√66
- Simplify the fraction. Divide the top and the bottom by any number that goes into every term.2√6
Check. Decimals: 4.899 before and 4.899 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.3√2
- Multiply the top and the bottom by √2. √2 ÷ √2 = 1, so multiplying by it does not change the value of the fraction. We choose √2 because √2 × √2 = 2, a whole number.3√2 × √2√2 = 3√2√2×√2
- Work out the top and the bottom. Top: multiply each term by √2. Bottom: √2 × √2 = 2.3√22
- The fraction cannot be simplified further. 3√22
Check. Decimals: 2.121 before and 2.121 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.1 + √3√2
- Multiply the top and the bottom by √2. √2 ÷ √2 = 1, so multiplying by it does not change the value of the fraction. We choose √2 because √2 × √2 = 2, a whole number.1 + √3√2 × √2√2 = (1 + √3)√2√2×√2
- Work out the top and the bottom. Top: multiply each term by √2. Bottom: √2 × √2 = 2.√2 + √62
Top: (1 + √3)√2 = √2 + √6 - The fraction cannot be simplified further. √2 + √62
Check. Decimals: 1.932 before and 1.932 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.2 + √5√5
- Multiply the top and the bottom by √5. √5 ÷ √5 = 1, so multiplying by it does not change the value of the fraction. We choose √5 because √5 × √5 = 5, a whole number.2 + √5√5 × √5√5 = (2 + √5)√5√5×√5
- Work out the top and the bottom. Top: multiply each term by √5. Bottom: √5 × √5 = 5.5 + 2√55
Top: (2 + √5)√5 = 5 + 2√5 - The fraction cannot be simplified further. 5 + 2√55
Check. Decimals: 1.894 before and 1.894 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.√2 + √6√2
- Multiply the top and the bottom by √2. √2 ÷ √2 = 1, so multiplying by it does not change the value of the fraction. We choose √2 because √2 × √2 = 2, a whole number.√2 + √6√2 × √2√2 = (√2 + √6)√2√2×√2
- Work out the top and the bottom. Top: multiply each term by √2. Bottom: √2 × √2 = 2.2 + 2√32
Top: (√2 + √6)√2 = 2 + 2√3 - Simplify the fraction. Divide the top and the bottom by any number that goes into every term.1 + √3
Check. Decimals: 2.732 before and 2.732 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the surd in the denominator. This makes the denominator a whole number.
- Say what is wrong with the fraction. A denominator (the bottom) containing a surd is not allowed in a fully simplified answer. Rationalising means rewriting the fraction so that the bottom is a whole number.3 + √5√5
- Multiply the top and the bottom by √5. √5 ÷ √5 = 1, so multiplying by it does not change the value of the fraction. We choose √5 because √5 × √5 = 5, a whole number.3 + √5√5 × √5√5 = (3 + √5)√5√5×√5
- Work out the top and the bottom. Top: multiply each term by √5. Bottom: √5 × √5 = 5.5 + 3√55
Top: (3 + √5)√5 = 5 + 3√5 - The fraction cannot be simplified further. 5 + 3√55
Check. Decimals: 2.342 before and 2.342 after. They agree. ✓
Watch out. You must multiply the top and the bottom. If you only multiply the bottom, you change the value of the fraction.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.13 + √5
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 3 + √5 conjugate: 3 − √5
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.1(3 + √5) × (3 − √5)(3 − √5)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(3 + √5)(3 − √5) = 32 − (√5)2 = 9 − 5 = 4
- Expand the top. Multiply the top by the conjugate, term by term.1 × (3 − √5) = 3 − √5
- Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.3 − √54 = 3 − √54
Check. Decimals: 0.191 before and 0.191 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.64 − √2
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 4 − √2 conjugate: 4 + √2
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.6(4 − √2) × (4 + √2)(4 + √2)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(4 − √2)(4 + √2) = 42 − (√2)2 = 16 − 2 = 14
- Expand the top. Multiply the top by the conjugate, term by term.6 × (4 + √2) = 24 + 6√2
- Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.24 + 6√214 = 12 + 3√27
Check. Decimals: 2.320 before and 2.320 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.21 + √3
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 1 + √3 conjugate: 1 − √3
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.2(1 + √3) × (1 − √3)(1 − √3)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(1 + √3)(1 − √3) = 12 − (√3)2 = 1 − 3 = −2
- Expand the top. Multiply the top by the conjugate, term by term.2 × (1 − √3) = 2 − 2√3
- Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.2 − 2√3(−2) = −1 + √3
Check. Decimals: 0.732 before and 0.732 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.12 − √3
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 2 − √3 conjugate: 2 + √3
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.1(2 − √3) × (2 + √3)(2 + √3)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(2 − √3)(2 + √3) = 22 − (√3)2 = 4 − 3 = 1
- Expand the top. Multiply the top by the conjugate, term by term.1 × (2 + √3) = 2 + √3
- Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.2 + √31 = 2 + √3
Check. Decimals: 3.732 before and 3.732 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.53 + √2
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 3 + √2 conjugate: 3 − √2
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.5(3 + √2) × (3 − √2)(3 − √2)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(3 + √2)(3 − √2) = 32 − (√2)2 = 9 − 2 = 7
- Expand the top. Multiply the top by the conjugate, term by term.5 × (3 − √2) = 15 − 5√2
- Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.15 − 5√27 = 15 − 5√27
Check. Decimals: 1.133 before and 1.133 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.2 + √31 + √3
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 1 + √3 conjugate: 1 − √3
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.2 + √3(1 + √3) × (1 − √3)(1 − √3)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(1 + √3)(1 − √3) = 12 − (√3)2 = 1 − 3 = −2
- Expand the top. Multiply the top by the conjugate, term by term.2 × 1 = 2
2 × (−√3) = −2√3
√3 × 1 = √3
√3 × (−√3) = −√9 = −3
Collected: −1 − √3 - Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.−1 − √3(−2) = 1 + √32
Check. Decimals: 1.366 before and 1.366 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.1 + √52 − √5
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 2 − √5 conjugate: 2 + √5
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.1 + √5(2 − √5) × (2 + √5)(2 + √5)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(2 − √5)(2 + √5) = 22 − (√5)2 = 4 − 5 = −1
- Expand the top. Multiply the top by the conjugate, term by term.1 × 2 = 2
1 × √5 = √5
√5 × 2 = 2√5
√5 × √5 = √25 = 5
Collected: 7 + 3√5 - Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.7 + 3√5(−1) = −7 − 3√5
Check. Decimals: -13.708 before and -13.708 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.3 + √24 + √2
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 4 + √2 conjugate: 4 − √2
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.3 + √2(4 + √2) × (4 − √2)(4 − √2)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(4 + √2)(4 − √2) = 42 − (√2)2 = 16 − 2 = 14
- Expand the top. Multiply the top by the conjugate, term by term.3 × 4 = 12
3 × (−√2) = −3√2
√2 × 4 = 4√2
√2 × (−√2) = −√4 = −2
Collected: 10 + √2 - Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.10 + √214 = 10 + √214
Check. Decimals: 0.815 before and 0.815 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.12 + √3
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 2 + √3 conjugate: 2 − √3
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.1(2 + √3) × (2 − √3)(2 − √3)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(2 + √3)(2 − √3) = 22 − (√3)2 = 4 − 3 = 1
- Expand the top. Multiply the top by the conjugate, term by term.1 × (2 − √3) = 2 − √3
- Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.2 − √31 = 2 − √3
Check. Decimals: 0.268 before and 0.268 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.43 − √5
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: 3 − √5 conjugate: 3 + √5
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.4(3 − √5) × (3 + √5)(3 + √5)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(3 − √5)(3 + √5) = 32 − (√5)2 = 9 − 5 = 4
- Expand the top. Multiply the top by the conjugate, term by term.4 × (3 + √5) = 12 + 4√5
- Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.12 + 4√54 = 3 + √5
Check. Decimals: 5.236 before and 5.236 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiply the top and bottom by the conjugate of the bottom. The bottom becomes a whole number (difference of two squares).
- Say what is wrong with the fraction. The bottom contains a surd. We need to rationalise it (make the bottom a whole number). Multiplying by the surd alone does not work here, because the bottom has two terms.1−1 + √3
- Write down the conjugate of the bottom. The conjugate is the same two terms with the sign between them changed. Multiplying a bracket by its conjugate gives a difference of two squares, so the surd disappears.bottom: −1 + √3 conjugate: −1 − √3
- Multiply top and bottom by the conjugate. Multiplying the top and the bottom by the same thing is multiplying by 1, so the value does not change.1(−1 + √3) × (−1 − √3)(−1 − √3)
- Expand the bottom using the difference of two squares. (p + q)(p − q) = p2 − q2, and the middle terms cancel.(−1 + √3)(−1 − √3) = -12 − (√3)2 = 1 − 3 = −2
- Expand the top. Multiply the top by the conjugate, term by term.1 × (−1 − √3) = −1 − √3
- Write the new fraction and simplify. Divide every term on the top by the number on the bottom if you can.−1 − √3(−2) = 1 + √32
Check. Decimals: 1.366 before and 1.366 after. They agree. ✓
Watch out. Change the sign between the two terms to get the conjugate, and multiply both the top and the bottom by it.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Multiplying only the bottom changes the value, so check Chloe's answer and then multiply the top and the bottom.
- Test Chloe's answer. Use decimals.1/√5 = 0.447 but 1/5 = 0.2
- Say what went wrong. To keep the value the same you must multiply the top and the bottom by √5.Chloe multiplied the bottom only.
- Do it correctly. √5 ÷ √5 = 1, so the value is unchanged.1√5 × √5√5 = √55
- Check. √5 ÷ 5 = 0.447, same as 1/√5. Correct answer: √55
Check. Both 1/√5 and √5/5 are 0.447. ✓
Watch out. Whatever you do to the bottom, do to the top.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Think of the multiplier as a fraction equal to 1.
- Write the multiplier. For example the multiplier is √2 on the top and √2 on the bottom.√2√2
- Work out its value. Any non-zero number divided by itself is 1.√2√2 = 1
- Use it. Multiplying by 1 leaves any value unchanged.1√2 × 1 = 1√2, which equals √22
Check. 1/√2 = 0.707 and √2/2 = 0.707. ✓
Watch out. This is the same idea as equivalent fractions.
Exercise 6R Exact answers with surds
Non-calculatorKey words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Use Pythagoras to get the square of the length, then take the square root and leave it as a simplified surd.
- Write Pythagoras' theorem. In a right-angled triangle the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides.c2 = a2 + b2
- Substitute the two shorter sides. c2 = 42 + 62 = 16 + 36 = 52
- Take the square root. Leave it as a surd, so that the answer is exact.side = √52
- Simplify the surd. The largest square factor of 52 is 4.√52 = √4 × 13 = 2√13
- Write the answer with units. This is the exact length.2√13 cm
Check. Square the answer: (2√13)2 = 52. As a decimal the length is about 7.21 cm. ✓
Watch out. Do not round if the question asks for an exact answer. Simplify the surd.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Use Pythagoras to get the square of the length, then take the square root and leave it as a simplified surd.
- Write Pythagoras' theorem. In a right-angled triangle the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides.c2 = a2 + b2
- Substitute the two shorter sides. c2 = 52 + 52 = 25 + 25 = 50
- Take the square root. Leave it as a surd, so that the answer is exact.side = √50
- Simplify the surd. The largest square factor of 50 is 25.√50 = √25 × 2 = 5√2
- Write the answer with units. This is the exact length.5√2 cm
Check. Square the answer: (5√2)2 = 50. As a decimal the length is about 7.07 cm. ✓
Watch out. Do not round if the question asks for an exact answer. Simplify the surd.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Use Pythagoras to get the square of the length, then take the square root and leave it as a simplified surd.
- Write Pythagoras' theorem. In a right-angled triangle the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides.c2 = a2 + b2
- Substitute the two shorter sides. c2 = 12 + 32 = 1 + 9 = 10
- Take the square root. Leave it as a surd, so that the answer is exact.side = √10
- Write the answer with units. This is the exact length.√10 cm
Check. Square the answer: (√10)2 = 10. As a decimal the length is about 3.16 cm. ✓
Watch out. Do not round if the question asks for an exact answer. Simplify the surd.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Use Pythagoras to get the square of the length, then take the square root and leave it as a simplified surd.
- Write Pythagoras' theorem for a shorter side. The longest side is the hypotenuse. To find a shorter side, subtract.a2 = c2 − b2
- Substitute. The hypotenuse is 7 and the other side is 3.a2 = 72 − 32 = 49 − 9 = 40
- Take the square root. Leave it as a surd, so that the answer is exact.side = √40
- Simplify the surd. The largest square factor of 40 is 4.√40 = √4 × 10 = 2√10
- Write the answer with units. This is the exact length.2√10 cm
Check. Square the answer: (2√10)2 = 40. As a decimal the length is about 6.32 cm. ✓
Watch out. Do not round if the question asks for an exact answer. Simplify the surd.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Use Pythagoras to get the square of the length, then take the square root and leave it as a simplified surd.
- Write Pythagoras' theorem. In a right-angled triangle the square on the longest side (the hypotenuse) equals the sum of the squares on the other two sides.c2 = a2 + b2
- Substitute the two shorter sides. c2 = 52 + 52 = 25 + 25 = 50
- Take the square root. Leave it as a surd, so that the answer is exact.side = √50
- Simplify the surd. The largest square factor of 50 is 25.√50 = √25 × 2 = 5√2
- Write the answer with units. This is the exact length.5√2 cm
Check. Square the answer: (5√2)2 = 50. As a decimal the length is about 7.07 cm. ✓
Watch out. Do not round if the question asks for an exact answer. Simplify the surd.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Use Pythagoras to get the square of the length, then take the square root and leave it as a simplified surd.
- Write Pythagoras' theorem for a shorter side. The longest side is the hypotenuse. To find a shorter side, subtract.a2 = c2 − b2
- Substitute. The hypotenuse is 2 and the other side is 1.a2 = 22 − 12 = 4 − 1 = 3
- Take the square root. Leave it as a surd, so that the answer is exact.side = √3
- Write the answer with units. This is the exact length.√3 cm
Check. Square the answer: (√3)2 = 3. As a decimal the length is about 1.73 cm. ✓
Watch out. Do not round if the question asks for an exact answer. Simplify the surd.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Area = ½ × base × height.
- Write the formula. The base is 2 cm and the height is √3 cm.A = ½ × 2 × √3
- Multiply. ½ × 2 = 1.A = 1 × √3 = √3 cm2
Check. Decimals: √3 = 1.732 cm2, and ½ × 2 × 1.732 = 1.732. ✓
Watch out. Use the exact height, not a rounded one.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Use area = ½ × base × height and simplify the surds.
- Write the area. A = ½ × √12 × √3
- Multiply the surds. √12 × √3 = √36.√12 × √3 = √36 = 6
- Halve. A = ½ × 6 = 3 cm2
Check. Decimals: 3.464 × 1.732 = 6.0, half is 3. ✓
Watch out. √a × √b = √(ab).
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Use Pythagoras and the rule (√a)2 = a.
- Write Pythagoras. c2 = (√6)2 + (√10)2
- Square each side. (√a)2 = a.c2 = 6 + 10 = 16
- Take the square root. c = √16 = 4 cm
Check. 4 × 4 = 16. ✓
Watch out. Squaring removes the surd.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Simplify every surd first. Then add or subtract the like surds (same number under the root).
- Simplify each surd on its own. Surds can only be added or subtracted when the number under the root is the same (like terms). Simplifying each surd often makes them match.√45 = √9 × 5 = 3√5
√20 = √4 × 5 = 2√5 - Rewrite the expression with the simplified surds. Keep the signs the same as before.3√5 + 2√5
- Collect the like surds. Like surds have the same number under the root. Add the numbers in front, just as you would add 3 apples and 5 apples. The root stays the same.√5 terms: 3 + 2 = 5, so 5√5
- Write the answer. 5√5
Check. Use decimals: the original is about 11.180 and the answer is about 11.180. They agree. ✓
Watch out. Only add or subtract surds that have the same number under the root. √2 + √3 is not √5.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Simplify every surd first. Then add or subtract the like surds (same number under the root).
- Simplify each surd on its own. Surds can only be added or subtracted when the number under the root is the same (like terms). Simplifying each surd often makes them match.3√20 = 3 × √4 × 5 = 3 × 2√5 = 6√5
√45 = √9 × 5 = 3√5 - Rewrite the expression with the simplified surds. Keep the signs the same as before.6√5 − 3√5
- Collect the like surds. Like surds have the same number under the root. Add the numbers in front, just as you would add 3 apples and 5 apples. The root stays the same.√5 terms: 6 − 3 = 3, so 3√5
- Write the answer. 3√5
Check. Use decimals: the original is about 6.708 and the answer is about 6.708. They agree. ✓
Watch out. Only add or subtract surds that have the same number under the root. √2 + √3 is not √5.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Simplify the first surd, divide, then add.
- Simplify √80. The largest square factor of 80 is 16.√80 = √(16 × 5) = 4√5
- Divide by 2. 4√5 ÷ 2 = 2√5
- Add. Like surds.2√5 + √5 = 3√5
Check. Decimals: 6.708 and 3√5 = 6.708. ✓
Watch out. Simplify first.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Choose two surds whose product is a square number.
- Choose two irrational numbers. √2 and √8 are both surds.√2 and √8
- Multiply. √a × √b = √(ab).√2 × √8 = √16
- Simplify. 16 is a square number.√16 = 4, which is rational.
Check. Another example: √3 × √3 = 3. ✓
Watch out. Any surd times itself is a whole number.
Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded. Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Use the area formula backwards: divide by π, take the square root, and simplify the surd.
- Write the formula for the area. A = πr2
- Put in the area and divide both sides by π. The area is 12π cm2. Dividing both sides by π leaves the number.12π = πr2, so r2 = 12
- Take the square root and leave it as a surd. 12 is not a perfect square, so keep it exact and simplify the surd: the largest square factor of 12 is 4.r = √12 = √(4 × 3) = 2√3
- Write the answer. 2√3 cm
Check. Put it back: π × (2√3)2 = π × 12 = 12π. ✓
Watch out. Leave the radius as a simplified surd for an exact answer.
Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded. Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Use C = 2πr with r = 2√3.
- Substitute. C = 2 × π × 2√3
- Multiply the numbers. 2 × 2 = 4.4π√3 cm
Check. Decimals: 4π√3 = 21.77 and 2π × 3.464 = 21.77. ✓
Watch out. Write the surd after π.
Key words. A surd is a square root that cannot be written as a whole number or fraction, such as √2 or √7 (√9 = 3 is not a surd). Surds are irrational: their decimals never end or repeat. We keep them as roots to give exact answers. The two rules to remember are √a × √b = √(ab) and √a × √a = a.
Method. Rationalise each fraction, then add.
- Rationalise the first fraction. Multiply by the conjugate (√2 − 1) on the top and bottom: the bottom becomes 2 − 1 = 1.1√2 + 1 = √2 − 11 = √2 − 1
- Rationalise the second fraction. The conjugate is √2 + 1 and the bottom is again 2 − 1 = 1.1√2 − 1 = √2 + 1
- Add. The −1 and +1 cancel.(√2 − 1) + (√2 + 1) = 2√2
Check. Decimals: 1/2.414 + 1/0.414 = 0.414 + 2.414 = 2.828 and 2√2 = 2.828. ✓
Watch out. The conjugate makes the bottom a whole number.