GCSE Maths: Surds: simplifying and adding (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)
Surds: simplifying and adding
Some square roots are whole numbers: √49 = 7. Most are not. √2 = 1.41421356… never ends and never repeats, so no fraction or terminating decimal equals it exactly. A root like this is called a surd (√2, √3, √5, √7…; but √9 = 3 is not a surd). Surds are irrational numbers. To give an exact answer we leave them as roots and keep them as simple as possible. Any decimal for a surd is only an approximation.
Two rules do almost all the work: √a × √b = √(ab), and √a × √a = a. Used backwards, the first rule lets us simplify a surd. √12 = √(4 × 3) = √4 × √3 = 2√3. To simplify √n: find the largest square number (4, 9, 16, 25, 36, 49, …) that divides into n, write n as (square) × (other), split the root, and take the root of the square number. A surd is fully simplified when the number under the root has no square factor. If there is a number in front, multiply it by the number you take out: 3√50 = 3 × 5√2 = 15√2.
Adding and subtracting surds works like collecting like terms. 3√2 + 5√2 = 8√2 (like 3 apples + 5 apples), but √2 + √3 cannot be combined. So simplify each surd first; they may then have the same number under the root. √12 + √27 = 2√3 + 3√3 = 5√3.
Key ideas
- A surd is a square root that is not a whole number (or fraction), such as √7. It is irrational.
- √a × √b = √(ab) and √a × √a = a.
- To simplify √n: use the largest square factor. √12 = √(4 × 3) = 2√3.
- Add and subtract only like surds (same number under the root). Simplify each surd 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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 12. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 12 ÷ 4 = 3 ✓ goes in exactly
9: 12 ÷ 9 = 1 remainder 3 ✗ does not go in exactly - Write 12 as 4 × 3. 4 is the largest square number that divides into 12. Use it so that the answer is fully simplified.12 = 4 × 3, so √12 = √4 × 3
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√4 × 3 = √4 × √3
- Work out √4. 4 is a square number, so its root is a whole number: 2 × 2 = 4.√4 = 2, so √4 × √3 = 2√3
- Write the answer. 2√3
Check. Square the answer: (2√3)2 = 22 × 3 = 4 × 3 = 12, and the original squared is (√12)2 = 12. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 50. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 50 ÷ 4 = 12 remainder 2 ✗ does not go in exactly
9: 50 ÷ 9 = 5 remainder 5 ✗ does not go in exactly
16: 50 ÷ 16 = 3 remainder 2 ✗ does not go in exactly
25: 50 ÷ 25 = 2 ✓ goes in exactly
36: 50 ÷ 36 = 1 remainder 14 ✗ does not go in exactly
49: 50 ÷ 49 = 1 remainder 1 ✗ does not go in exactly - Write 50 as 25 × 2. 25 is the largest square number that divides into 50. Use it so that the answer is fully simplified.50 = 25 × 2, so √50 = √25 × 2
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√25 × 2 = √25 × √2
- Work out √25. 25 is a square number, so its root is a whole number: 5 × 5 = 25.√25 = 5, so √25 × √2 = 5√2
- Multiply by the number that was already in front. The original expression was 3 lots of √50, so multiply the 5 by 3.3 × 5√2 = 15√2
- Write the answer. 15√2
Check. Square the answer: (15√2)2 = 152 × 2 = 225 × 2 = 450, and the original squared is (3√50)2 = 450. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Look for the largest square number that divides into the number under the root.
- List the square numbers and see which ones divide into 35. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 35 ÷ 4 = 8 remainder 3 ✗ does not go in exactly
9: 35 ÷ 9 = 3 remainder 8 ✗ does not go in exactly
16: 35 ÷ 16 = 2 remainder 3 ✗ does not go in exactly
25: 35 ÷ 25 = 1 remainder 10 ✗ does not go in exactly - Decide. No square number except 1 divides into 35, so √35 is already in its simplest form.√35 cannot be simplified.
- Write the answer. A surd is in its simplest form when the number under the root has no square factor (other than 1).√35
Check. Check: no square number between 2 and 35 divides exactly into 35. ✓
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.√12 = √4 × 3 = 2√3
√27 = √9 × 3 = 3√3 - Rewrite the expression with the simplified surds. Keep the signs the same as before.2√3 + 3√3
- 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.√3 terms: 2 + 3 = 5, so 5√3
- Write the answer. 5√3
Check. Use decimals: the original is about 8.660 and the answer is about 8.660. 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.5√2: already simplest
√8 = √4 × 2 = 2√2
2√18 = 2 × √9 × 2 = 2 × 3√2 = 6√2 - Rewrite the expression with the simplified surds. Keep the signs the same as before.5√2 − 2√2 + 6√2
- 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.√2 terms: 5 − 2 + 6 = 9, so 9√2
- Write the answer. 9√2
Check. Use decimals: the original is about 12.728 and the answer is about 12.728. 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. Take the square root of the top and the bottom separately, then simplify each.
- Split the root of a fraction. √(a/b) = √a ÷ √b.√29 = √2√9
- Simplify the top and the bottom. Take square factors out of each root.√2 = √2, √9 = 3
- Write the answer as a fraction. Cancel any common factor.√23
Check. Decimals: √(2/9) = 0.4714 and the answer is 0.4714. ✓
Watch out. Only the bottom can have its surd removed later by rationalising (section 6.9).
Watch out
√2 + √3 is not √5, and √(9 + 16) is not √9 + √16. You can only add surds with the same number under the root. Also use the largest square factor, or check afterwards that nothing under the root can still be simplified.Exercise 6O Simplifying and adding 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. Check whether the number under the root is a square number.
- Work out the root. If the root is a whole number, it is not a surd.√16 = 4, because 4 × 4 = 16
- Decide. A surd is a root that is not a whole number or fraction.4 is a whole number, so √16 is not a surd.
Check. 42 = 16. ✓
Watch out. √16 = 4 is a normal number.
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. Check whether the number under the root is a square number.
- Look for a whole-number root. 22 = 4 and 32 = 9, and 7 lies between them.4 < 7 < 9
- Decide. The root is between two whole numbers, so it is not a whole number. Its decimal never ends or repeats, so it is irrational.√7 is a surd.
Check. A calculator gives √7 = 2.64575…, not a whole number. ✓
Watch out. A surd can sometimes be simplified (such as √12 = 2√3) but it is still a 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. Check whether the number under the root is a square number.
- Look for a whole-number root. 72 = 49 and 82 = 64, and 50 lies between them.49 < 50 < 64
- Decide. The root is between two whole numbers, so it is not a whole number. Its decimal never ends or repeats, so it is irrational.√50 is a surd.
Check. A calculator gives √50 = 7.07107…, not a whole number. ✓
Watch out. A surd can sometimes be simplified (such as √12 = 2√3) but it is still a 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. Check whether the number under the root is a square number.
- Work out the root. If the root is a whole number, it is not a surd.√81 = 9, because 9 × 9 = 81
- Decide. A surd is a root that is not a whole number or fraction.9 is a whole number, so √81 is not a surd.
Check. 92 = 81. ✓
Watch out. √16 = 4 is a normal number.
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. Check whether the number under the root is a square number.
- Look for a whole-number root. 12 = 1 and 22 = 4, and 3 lies between them.1 < 3 < 4
- Decide. The root is between two whole numbers, so it is not a whole number. Its decimal never ends or repeats, so it is irrational.√3 is a surd.
Check. A calculator gives √3 = 1.73205…, not a whole number. ✓
Watch out. A surd can sometimes be simplified (such as √12 = 2√3) but it is still a 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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 12. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 12 ÷ 4 = 3 ✓ goes in exactly
9: 12 ÷ 9 = 1 remainder 3 ✗ does not go in exactly - Write 12 as 4 × 3. 4 is the largest square number that divides into 12. Use it so that the answer is fully simplified.12 = 4 × 3, so √12 = √4 × 3
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√4 × 3 = √4 × √3
- Work out √4. 4 is a square number, so its root is a whole number: 2 × 2 = 4.√4 = 2, so √4 × √3 = 2√3
- Write the answer. 2√3
Check. Square the answer: (2√3)2 = 22 × 3 = 4 × 3 = 12, and the original squared is (√12)2 = 12. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 18. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 18 ÷ 4 = 4 remainder 2 ✗ does not go in exactly
9: 18 ÷ 9 = 2 ✓ goes in exactly
16: 18 ÷ 16 = 1 remainder 2 ✗ does not go in exactly - Write 18 as 9 × 2. 9 is the largest square number that divides into 18. Use it so that the answer is fully simplified.18 = 9 × 2, so √18 = √9 × 2
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√9 × 2 = √9 × √2
- Work out √9. 9 is a square number, so its root is a whole number: 3 × 3 = 9.√9 = 3, so √9 × √2 = 3√2
- Write the answer. 3√2
Check. Square the answer: (3√2)2 = 32 × 2 = 9 × 2 = 18, and the original squared is (√18)2 = 18. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 20. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 20 ÷ 4 = 5 ✓ goes in exactly
9: 20 ÷ 9 = 2 remainder 2 ✗ does not go in exactly
16: 20 ÷ 16 = 1 remainder 4 ✗ does not go in exactly - Write 20 as 4 × 5. 4 is the largest square number that divides into 20. Use it so that the answer is fully simplified.20 = 4 × 5, so √20 = √4 × 5
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√4 × 5 = √4 × √5
- Work out √4. 4 is a square number, so its root is a whole number: 2 × 2 = 4.√4 = 2, so √4 × √5 = 2√5
- Write the answer. 2√5
Check. Square the answer: (2√5)2 = 22 × 5 = 4 × 5 = 20, and the original squared is (√20)2 = 20. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 45. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 45 ÷ 4 = 11 remainder 1 ✗ does not go in exactly
9: 45 ÷ 9 = 5 ✓ goes in exactly
16: 45 ÷ 16 = 2 remainder 13 ✗ does not go in exactly
25: 45 ÷ 25 = 1 remainder 20 ✗ does not go in exactly
36: 45 ÷ 36 = 1 remainder 9 ✗ does not go in exactly - Write 45 as 9 × 5. 9 is the largest square number that divides into 45. Use it so that the answer is fully simplified.45 = 9 × 5, so √45 = √9 × 5
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√9 × 5 = √9 × √5
- Work out √9. 9 is a square number, so its root is a whole number: 3 × 3 = 9.√9 = 3, so √9 × √5 = 3√5
- Write the answer. 3√5
Check. Square the answer: (3√5)2 = 32 × 5 = 9 × 5 = 45, and the original squared is (√45)2 = 45. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 75. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 75 ÷ 4 = 18 remainder 3 ✗ does not go in exactly
9: 75 ÷ 9 = 8 remainder 3 ✗ does not go in exactly
16: 75 ÷ 16 = 4 remainder 11 ✗ does not go in exactly
25: 75 ÷ 25 = 3 ✓ goes in exactly
36: 75 ÷ 36 = 2 remainder 3 ✗ does not go in exactly
49: 75 ÷ 49 = 1 remainder 26 ✗ does not go in exactly
64: 75 ÷ 64 = 1 remainder 11 ✗ does not go in exactly - Write 75 as 25 × 3. 25 is the largest square number that divides into 75. Use it so that the answer is fully simplified.75 = 25 × 3, so √75 = √25 × 3
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√25 × 3 = √25 × √3
- Work out √25. 25 is a square number, so its root is a whole number: 5 × 5 = 25.√25 = 5, so √25 × √3 = 5√3
- Write the answer. 5√3
Check. Square the answer: (5√3)2 = 52 × 3 = 25 × 3 = 75, and the original squared is (√75)2 = 75. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 72. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 72 ÷ 4 = 18 ✓ goes in exactly
9: 72 ÷ 9 = 8 ✓ goes in exactly
16: 72 ÷ 16 = 4 remainder 8 ✗ does not go in exactly
25: 72 ÷ 25 = 2 remainder 22 ✗ does not go in exactly
36: 72 ÷ 36 = 2 ✓ goes in exactly
49: 72 ÷ 49 = 1 remainder 23 ✗ does not go in exactly
64: 72 ÷ 64 = 1 remainder 8 ✗ does not go in exactly - Write 72 as 36 × 2. 36 is the largest square number that divides into 72. Use it so that the answer is fully simplified.72 = 36 × 2, so √72 = √36 × 2
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√36 × 2 = √36 × √2
- Work out √36. 36 is a square number, so its root is a whole number: 6 × 6 = 36.√36 = 6, so √36 × √2 = 6√2
- Write the answer. 6√2
Check. Square the answer: (6√2)2 = 62 × 2 = 36 × 2 = 72, and the original squared is (√72)2 = 72. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 98. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 98 ÷ 4 = 24 remainder 2 ✗ does not go in exactly
9: 98 ÷ 9 = 10 remainder 8 ✗ does not go in exactly
16: 98 ÷ 16 = 6 remainder 2 ✗ does not go in exactly
25: 98 ÷ 25 = 3 remainder 23 ✗ does not go in exactly
36: 98 ÷ 36 = 2 remainder 26 ✗ does not go in exactly
49: 98 ÷ 49 = 2 ✓ goes in exactly
64: 98 ÷ 64 = 1 remainder 34 ✗ does not go in exactly
81: 98 ÷ 81 = 1 remainder 17 ✗ does not go in exactly - Write 98 as 49 × 2. 49 is the largest square number that divides into 98. Use it so that the answer is fully simplified.98 = 49 × 2, so √98 = √49 × 2
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√49 × 2 = √49 × √2
- Work out √49. 49 is a square number, so its root is a whole number: 7 × 7 = 49.√49 = 7, so √49 × √2 = 7√2
- Write the answer. 7√2
Check. Square the answer: (7√2)2 = 72 × 2 = 49 × 2 = 98, and the original squared is (√98)2 = 98. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 200. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 200 ÷ 4 = 50 ✓ goes in exactly
9: 200 ÷ 9 = 22 remainder 2 ✗ does not go in exactly
16: 200 ÷ 16 = 12 remainder 8 ✗ does not go in exactly
25: 200 ÷ 25 = 8 ✓ goes in exactly
36: 200 ÷ 36 = 5 remainder 20 ✗ does not go in exactly
49: 200 ÷ 49 = 4 remainder 4 ✗ does not go in exactly
64: 200 ÷ 64 = 3 remainder 8 ✗ does not go in exactly
81: 200 ÷ 81 = 2 remainder 38 ✗ does not go in exactly
100: 200 ÷ 100 = 2 ✓ goes in exactly
121: 200 ÷ 121 = 1 remainder 79 ✗ does not go in exactly
144: 200 ÷ 144 = 1 remainder 56 ✗ does not go in exactly
169: 200 ÷ 169 = 1 remainder 31 ✗ does not go in exactly
196: 200 ÷ 196 = 1 remainder 4 ✗ does not go in exactly - Write 200 as 100 × 2. 100 is the largest square number that divides into 200. Use it so that the answer is fully simplified.200 = 100 × 2, so √200 = √100 × 2
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√100 × 2 = √100 × √2
- Work out √100. 100 is a square number, so its root is a whole number: 10 × 10 = 100.√100 = 10, so √100 × √2 = 10√2
- Write the answer. 10√2
Check. Square the answer: (10√2)2 = 102 × 2 = 100 × 2 = 200, and the original squared is (√200)2 = 200. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 8. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 8 ÷ 4 = 2 ✓ goes in exactly
- Write 8 as 4 × 2. 4 is the largest square number that divides into 8. Use it so that the answer is fully simplified.8 = 4 × 2, so √8 = √4 × 2
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√4 × 2 = √4 × √2
- Work out √4. 4 is a square number, so its root is a whole number: 2 × 2 = 4.√4 = 2, so √4 × √2 = 2√2
- Multiply by the number that was already in front. The original expression was 3 lots of √8, so multiply the 2 by 3.3 × 2√2 = 6√2
- Write the answer. 6√2
Check. Square the answer: (6√2)2 = 62 × 2 = 36 × 2 = 72, and the original squared is (3√8)2 = 72. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 27. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 27 ÷ 4 = 6 remainder 3 ✗ does not go in exactly
9: 27 ÷ 9 = 3 ✓ goes in exactly
16: 27 ÷ 16 = 1 remainder 11 ✗ does not go in exactly
25: 27 ÷ 25 = 1 remainder 2 ✗ does not go in exactly - Write 27 as 9 × 3. 9 is the largest square number that divides into 27. Use it so that the answer is fully simplified.27 = 9 × 3, so √27 = √9 × 3
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√9 × 3 = √9 × √3
- Work out √9. 9 is a square number, so its root is a whole number: 3 × 3 = 9.√9 = 3, so √9 × √3 = 3√3
- Multiply by the number that was already in front. The original expression was 2 lots of √27, so multiply the 3 by 2.2 × 3√3 = 6√3
- Write the answer. 6√3
Check. Square the answer: (6√3)2 = 62 × 3 = 36 × 3 = 108, and the original squared is (2√27)2 = 108. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 12. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 12 ÷ 4 = 3 ✓ goes in exactly
9: 12 ÷ 9 = 1 remainder 3 ✗ does not go in exactly - Write 12 as 4 × 3. 4 is the largest square number that divides into 12. Use it so that the answer is fully simplified.12 = 4 × 3, so √12 = √4 × 3
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√4 × 3 = √4 × √3
- Work out √4. 4 is a square number, so its root is a whole number: 2 × 2 = 4.√4 = 2, so √4 × √3 = 2√3
- Multiply by the number that was already in front. The original expression was 5 lots of √12, so multiply the 2 by 5.5 × 2√3 = 10√3
- Write the answer. 10√3
Check. Square the answer: (10√3)2 = 102 × 3 = 100 × 3 = 300, and the original squared is (5√12)2 = 300. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 50. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 50 ÷ 4 = 12 remainder 2 ✗ does not go in exactly
9: 50 ÷ 9 = 5 remainder 5 ✗ does not go in exactly
16: 50 ÷ 16 = 3 remainder 2 ✗ does not go in exactly
25: 50 ÷ 25 = 2 ✓ goes in exactly
36: 50 ÷ 36 = 1 remainder 14 ✗ does not go in exactly
49: 50 ÷ 49 = 1 remainder 1 ✗ does not go in exactly - Write 50 as 25 × 2. 25 is the largest square number that divides into 50. Use it so that the answer is fully simplified.50 = 25 × 2, so √50 = √25 × 2
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√25 × 2 = √25 × √2
- Work out √25. 25 is a square number, so its root is a whole number: 5 × 5 = 25.√25 = 5, so √25 × √2 = 5√2
- Multiply by the number that was already in front. The original expression was 4 lots of √50, so multiply the 5 by 4.4 × 5√2 = 20√2
- Write the answer. 20√2
Check. Square the answer: (20√2)2 = 202 × 2 = 400 × 2 = 800, and the original squared is (4√50)2 = 800. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 18. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 18 ÷ 4 = 4 remainder 2 ✗ does not go in exactly
9: 18 ÷ 9 = 2 ✓ goes in exactly
16: 18 ÷ 16 = 1 remainder 2 ✗ does not go in exactly - Write 18 as 9 × 2. 9 is the largest square number that divides into 18. Use it so that the answer is fully simplified.18 = 9 × 2, so √18 = √9 × 2
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√9 × 2 = √9 × √2
- Work out √9. 9 is a square number, so its root is a whole number: 3 × 3 = 9.√9 = 3, so √9 × √2 = 3√2
- Multiply by the number that was already in front. The original expression was 7 lots of √18, so multiply the 3 by 7.7 × 3√2 = 21√2
- Write the answer. 21√2
Check. Square the answer: (21√2)2 = 212 × 2 = 441 × 2 = 882, and the original squared is (7√18)2 = 882. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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.√12 = √4 × 3 = 2√3
√27 = √9 × 3 = 3√3 - Rewrite the expression with the simplified surds. Keep the signs the same as before.2√3 + 3√3
- 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.√3 terms: 2 + 3 = 5, so 5√3
- Write the answer. 5√3
Check. Use decimals: the original is about 8.660 and the answer is about 8.660. 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.5√2: already simplest
√8 = √4 × 2 = 2√2 - Rewrite the expression with the simplified surds. Keep the signs the same as before.5√2 − 2√2
- 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.√2 terms: 5 − 2 = 3, so 3√2
- Write the answer. 3√2
Check. Use decimals: the original is about 4.243 and the answer is about 4.243. 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√18 = 3 × √9 × 2 = 3 × 3√2 = 9√2
√32 = √16 × 2 = 4√2 - Rewrite the expression with the simplified surds. Keep the signs the same as before.9√2 + 4√2
- 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.√2 terms: 9 + 4 = 13, so 13√2
- Write the answer. 13√2
Check. Use decimals: the original is about 18.385 and the answer is about 18.385. 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.2√20 = 2 × √4 × 5 = 2 × 2√5 = 4√5
√45 = √9 × 5 = 3√5 - Rewrite the expression with the simplified surds. Keep the signs the same as before.4√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: 4 − 3 = 1, so √5
- Write the answer. √5
Check. Use decimals: the original is about 2.236 and the answer is about 2.236. 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.√75 = √25 × 3 = 5√3
√12 = √4 × 3 = 2√3
√3: already simplest - Rewrite the expression with the simplified surds. Keep the signs the same as before.5√3 + 2√3 − √3
- 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.√3 terms: 5 + 2 − 1 = 6, so 6√3
- Write the answer. 6√3
Check. Use decimals: the original is about 10.392 and the answer is about 10.392. 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.√12 = √4 × 3 = 2√3
√5: already simplest - Rewrite the expression with the simplified surds. Keep the signs the same as before.2√3 − √5
- Write the answer. If the surds are different (like √2 and √3) they cannot be combined, so the answer stays as it is.2√3 − √5
Check. Use decimals: the original is about 1.228 and the answer is about 1.228. 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. Take the square root of the top and the bottom separately, then simplify each.
- Split the root of a fraction. √(a/b) = √a ÷ √b.√29 = √2√9
- Simplify the top and the bottom. Take square factors out of each root.√2 = √2, √9 = 3
- Write the answer as a fraction. Cancel any common factor.√23
Check. Decimals: √(2/9) = 0.4714 and the answer is 0.4714. ✓
Watch out. Only the bottom can have its surd removed later by rationalising (section 6.9).
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. Take the square root of the top and the bottom separately, then simplify each.
- Split the root of a fraction. √(a/b) = √a ÷ √b.√825 = √8√25
- Simplify the top and the bottom. Take square factors out of each root.√8 = 2√2, √25 = 5
- Write the answer as a fraction. Cancel any common factor.2√25
Check. Decimals: √(8/25) = 0.5657 and the answer is 0.5657. ✓
Watch out. Only the bottom can have its surd removed later by rationalising (section 6.9).
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. Take the square root of the top and the bottom separately, then simplify each.
- Split the root of a fraction. √(a/b) = √a ÷ √b.√316 = √3√16
- Simplify the top and the bottom. Take square factors out of each root.√3 = √3, √16 = 4
- Write the answer as a fraction. Cancel any common factor.√34
Check. Decimals: √(3/16) = 0.4330 and the answer is 0.4330. ✓
Watch out. Only the bottom can have its surd removed later by rationalising (section 6.9).
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. Take the square root of the top and the bottom separately, then simplify each.
- Split the root of a fraction. √(a/b) = √a ÷ √b.√1249 = √12√49
- Simplify the top and the bottom. Take square factors out of each root.√12 = 2√3, √49 = 7
- Write the answer as a fraction. Cancel any common factor.2√37
Check. Decimals: √(12/49) = 0.4949 and the answer is 0.4949. ✓
Watch out. Only the bottom can have its surd removed later by rationalising (section 6.9).
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. Perimeter = 2 × (length + width). Simplify each side, then add.
- Simplify each side. Take square factors out.√18 = √(9 × 2) = 3√2
√8 = √(4 × 2) = 2√2 - Add the length and width. The surds are like surds, so add the numbers in front.3√2 + 2√2 = 5√2
- Double it. A rectangle has two lengths and two widths.2 × 5√2 = 10√2 cm
Check. Decimals: √18 + √8 = 7.071, doubled 14.142; and 10√2 = 14.142. ✓
Watch out. Simplify first so the surds match.
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.
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 = 32 + 32 = 9 + 9 = 18
- Take the square root. Leave it as a surd, so that the answer is exact.side = √18
- Simplify the surd. The largest square factor of 18 is 9.√18 = √9 × 2 = 3√2
- Write the answer with units. This is the exact length.3√2 cm
Check. Square the answer: (3√2)2 = 18. As a decimal the length is about 4.24 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. Find the largest square number that is a factor of the number under the root. Split the root into a product, and take the root of the square number.
- List the square numbers and see which ones divide into 72. To simplify a surd we look for a square number factor: 4, 9, 16, 25, 36, 49, 64, 81, 100… The biggest one that divides exactly is the best one to use.4: 72 ÷ 4 = 18 ✓ goes in exactly
9: 72 ÷ 9 = 8 ✓ goes in exactly
16: 72 ÷ 16 = 4 remainder 8 ✗ does not go in exactly
25: 72 ÷ 25 = 2 remainder 22 ✗ does not go in exactly
36: 72 ÷ 36 = 2 ✓ goes in exactly
49: 72 ÷ 49 = 1 remainder 23 ✗ does not go in exactly
64: 72 ÷ 64 = 1 remainder 8 ✗ does not go in exactly - Write 72 as 36 × 2. 36 is the largest square number that divides into 72. Use it so that the answer is fully simplified.72 = 36 × 2, so √72 = √36 × 2
- Split the root into two roots. The rule is √(a × b) = √a × √b. This works because squaring each side gives the same number.√36 × 2 = √36 × √2
- Work out √36. 36 is a square number, so its root is a whole number: 6 × 6 = 36.√36 = 6, so √36 × √2 = 6√2
- Write the answer. 6√2
Check. Square the answer: (6√2)2 = 62 × 2 = 36 × 2 = 72, and the original squared is (√72)2 = 72. ✓
Watch out. If you choose a smaller square factor you must carry on simplifying. For example √72 = √(4 × 18) = 2√18, and √18 simplifies again to 3√2. Always check whether the surd left can be simplified more.
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. Work out each side as a decimal using a calculator and compare.
- Left-hand side. Use a calculator for each root and add.√2 + √3 = 1.414 + 1.732 = 3.146
- Right-hand side. √5 = 2.236
- Compare. The values are different.No, they are not equal. Surds with different numbers under the root cannot be added into one root.
Check. A different test: square both sides. (√2 + √3)2 = 5 + 2√6, which is not 5. ✓
Watch out. You can only combine like surds.
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. Check whether the number under the root is a square number.
- Look for a whole-number root. 42 = 16 and 52 = 25, and 18 lies between them.16 < 18 < 25
- Decide. The root is between two whole numbers, so it is not a whole number. Its decimal never ends or repeats, so it is irrational.√18 is a surd.
Check. A calculator gives √18 = 4.24264…, not a whole number. ✓
Watch out. A surd can sometimes be simplified (such as √12 = 2√3) but it is still a 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. Square each expression using (√a)2 = a.
- Square √12. A square root squared gives the number back.(√12)2 = 12
- Square 2√3. Square the 2 and square the √3 separately.(2√3)2 = 22 × (√3)2 = 4 × 3
- Compare. 4 × 3 = 12, so both are 12. This also shows √12 = 2√3.
Check. Decimals: √12 = 3.464 and 2√3 = 2 × 1.732 = 3.464. ✓
Watch out. Square the number in front as well.
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. Write each as a square root, or square each number, so they can be compared.
- Square each number. Squaring keeps the order for positive numbers.(√20)2 = 20
4.52 = 20.25
(3√2)2 = 9 × 2 = 18 - Order the squares. Smallest first: 18, 20, 20.25.18 < 20 < 20.25
- Write the original numbers in that order. 3√2, √20, 4.5
Check. Decimals: 3√2 = 4.243, √20 = 4.472, 4.5. ✓
Watch out. Compare the squares, not the roots, if you have no calculator.