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This page sets out the method for surds in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Surds topic page.
When you see with not a square number, look for the largest square number that divides into .
When the denominator is or , multiply top and bottom by the same expression with the middle sign changed.
A surd simplifies when a square number divides the value under the root.
√8 and √18 are both multiples of √2, so they can be added.
To rationalise 1/√3, multiply the top and bottom by √3.
5/√5 becomes √5 because 5 and the new denominator share a factor.
(√3 + 1)² is 3 + 2√3 + 1, which is 4 + 2√3.
Leave the surd exact. Do not replace √3 with a rounded decimal.
For a denominator such as 2 + √3, multiply top and bottom by its conjugate, 2 − √3 (the same numbers with the middle sign changed). The bottom becomes (2 + √3)(2 − √3) = 4 − 3 = 1, so 1/(2 + √3) = 2 − √3.
The middle terms cancel in (a + √b)(a − √b), so the answer is a² − b with no surd left. Remember to multiply the top by the same bracket and to expand it carefully.
Look for the largest square that divides the number under the root. The squares to try are 4, 9, 16, 25, 36, 49, 64, 100. For √(72), 36 divides 72 and 36 = 6^2. Then √(72) = √(36 × 2) = √(36) × √(2) = 6√(2). Using 4 also works, but it leaves a job unfinished: √(72) = 2√(18) and √(18) still has a square factor.
√(48) = √(16 × 3) = 4√(3). √(50) = √(25 × 2) = 5√(2). √(18) = √(9 × 2) = 3√(2). √(12) = √(4 × 3) = 2√(3). Check by squaring the answer. (6√(2))^2 = 36 × 2 = 72. If the square does not return the original number, the factor outside the root is wrong.
A root of a product splits. A root of a sum does not. √(36 × 2) = 6√(2), but √(36 + 2) is √(38), which is a different number. When you simplify, the square factor multiplies the rest. It is not added to it. Write the product under one root before you split.
Method: 1. Find the largest square that divides the number under the root. 2. Split the root into that square times what is left. 3. Take the root of the square and leave the other factor inside the root.
√(50) + √(18) = 5√(2) + 3√(2) = 8√(2). The root is the same, so the coefficients add: 5 + 3 = 8. √(12) + √(27) = 2√(3) + 3√(3) = 5√(3). Adding under one root, √(50 + 18) = √(68), is a different value. Collect like surds the way you collect like terms in algebra. √(2) and √(3) are different terms and stay separate.
A squared bracket needs three products. (2 + √(3))^2 = 2^2 + 2 × 2 × √(3) + (√(3))^2 = 4 + 4√(3) + 3 = 7 + 4√(3). The middle term is twice the product of the two terms inside the bracket. Missing it is the usual error. (√(3))^2 is 3, not √(3). A squared surd is the number under the root, provided the root is the positive square root.
The difference of two squares is the clean version of that expansion. (√(3) + 1)(√(3) - 1) = (√(3))^2 - 1^2 = 3 - 1 = 2. The cross terms +√(3) and -√(3) cancel. This identity is the reason a conjugate rationalises a two-term denominator: the product is a whole number.
Method: 1. Simplify each surd first so like roots show up. 2. Add or subtract only the coefficients of the same root. 3. Expand a bracket by multiplying each term, then collect like surds.
dfrac1√(5) still has a surd underneath. Multiply the top and the bottom by √(5). The bottom becomes √(5) × √(5) = 5. The top becomes √(5). So dfrac1√(5) = dfrac√(5)5. The value has not changed, because you multiplied by √(5)/√(5), which is 1. The denominator is now a whole number.
A number on top is carried through. dfrac6√(3) = dfrac6√(3)3 = 2√(3). After the multiply, cancel the whole numbers. 6/3 = 2, and the √(3) stays on top. Leaving the answer as 6√(3)/3 is correct but not finished. An answer of 6√(3) has forgotten the division by 3.
√(48) and 6/√(3) are different simplifications that both end in multiples of √(3). √(48) = 4√(3) and 6/√(3) = 2√(3). Do not cancel the 3 under the root with the 3 on the bottom straight away. They only link after you have written √(3) × √(3) = 3.
Method: 1. Multiply the top and the bottom by the surd in the denominator. 2. The bottom becomes a whole number. 3. Simplify the fraction if a common factor remains.
If the denominator is √(3) + 1, multiply top and bottom by √(3) - 1. The bottom becomes 3 - 1 = 2. The top is whatever you started with, multiplied by √(3) - 1. Then cancel any common factor of the whole top and the whole bottom. The sign you switch is the one in the middle of the denominator. The conjugate of 2 + √(3) is 2 - √(3).
Simplify a root in the denominator before you choose the conjugate, if it makes the terms smaller. √(12) + 2 = 2√(3) + 2 = 2(√(3) + 1). A fraction with that denominator can be reduced by the factor of 2 first. Then the conjugate of √(3) + 1 does the rest. Doing the conjugate on 2√(3) + 2 also works. It is more writing, and the bottom becomes (2√(3))^2 - 2^2 = 12 - 4 = 8.
The finished answer should have an integer denominator, and like surds collected. 2√(3) - 2 is finished. dfrac4√(3) - 42 is one cancel away from finished. Divide each term by 2. Do not leave a surd downstairs, and do not convert 2√(3) - 2 into a decimal. The exact form is the answer.
Method: 1. Multiply the top and the bottom by the conjugate of the denominator. 2. The bottom becomes a difference of two squares, with no surd left. 3. Expand the top, then simplify.
Geometry questions give lengths as surds and ask for an exact answer, "in the form a√(b)" or "in its simplest form". Squaring a surd removes the root: (√(7))^2 = 7 and (3√(2))^2 = 18. So in a right-angled triangle with shorter sides 3√(2) and √(14), the hypotenuse squared is 18 + 14 = 32, and the hypotenuse is √(32) = 4√(2). Always square the whole length, including any number in front of the root.
For an area, multiply the lengths and then simplify the root. The area of that triangle is tfrac12 × 3√(2) × √(14). Multiply the surds, √(2) × √(14) = √(28) = 2√(7), and then the numbers, tfrac12 × 3 × 2 = 3. So the area is 3√(7). Keep the working exact until the end. A decimal such as 7.94 loses the accuracy mark in an exact-answer question.
Method: 1. Square each length, remembering to square the number in front of any root. 2. Add or subtract the squares to find the missing side squared. 3. Take the square root and simplify using the largest square factor. 4. For area, multiply the surds under one root, simplify, and then multiply the numbers.
Tip: If a question says "exact", leave the answer as a surd and do not use a decimal.
To multiply surds, multiply the numbers outside the roots and multiply the numbers under the roots: 2√(3) × 5√(2) = 10√(6). Remember that √(a) × √(a) = a. To expand (3 + √(5))(2 - √(5)), multiply each term in the first bracket by each term in the second, which gives 6 - 3√(5) + 2√(5) - 5. Then collect the whole numbers and the surd terms separately to get 1 - √(5).
Two patterns appear often. A squared bracket such as (2√(3) + 1)^2 must be written as two brackets and expanded in full, which gives 12 + 4√(3) + 1 = 13 + 4√(3). The difference of two squares (a + √(b))(a - √(b)) = a^2 - b has no surd term left, so it is a whole number. If a question asks you to show that a result is rational, this is usually the pattern it wants.
Method: 1. Write every term-by-term product of the two brackets. 2. Replace √(a) × √(a) with a, and combine other surds under one root. 3. Collect whole numbers together and surd terms together. 4. Give the answer in the form asked for, such as a + b√(c).
When the denominator is a sum or difference involving a surd, multiply the top and bottom by the conjugate, which is the same expression with the sign in the middle changed. For dfrac10√(6) - 2 the conjugate is √(6) + 2. The new denominator is (√(6) - 2)(√(6) + 2) = 6 - 4 = 2, a whole number, and the numerator is 10(√(6) + 2). Dividing by 2 gives 5(√(6) + 2) = 10 + 5√(6).
Questions often end with "in the form a + b√(c)" or "show that it can be written as dfraca + b√(c)d". Check that your last line has exactly that shape, with the whole number and the surd term separated. For a show-that, write the conjugate multiplication, the new denominator, the expanded numerator and the final simplified fraction. A decimal check helps: with √(6) ≈ 2.4495, both 10/(2.4495 - 2) and 10 + 5 × 2.4495 are about 22.25.
Method: 1. Write the conjugate of the denominator by changing the sign between the two terms. 2. Multiply the top and the bottom by the conjugate. 3. Expand the denominator with the difference of two squares and expand the numerator in full. 4. Simplify and write the answer in the form asked for.
You cannot use a calculator, but you can still check a surd answer. Use √(2) ≈ 1.414, √(3) ≈ 1.732, √(5) ≈ 2.236 and √(7) ≈ 2.646. Put the original expression and your simplified form into approximate decimals and see whether they agree to two decimal places. This catches errors such as √(18) + √(8) = √(26), because 4.24 + 2.83 = 7.07 but √(26) ≈ 5.10.
Explanation questions ask why, for example, √(2) + √(3) is not √(5). A good reason is a counter-check: √(2) + √(3) is about 3.15 and √(5) is about 2.24. For find-the-mistake questions, read each line and find the first line that is not equal to the one before. Name it, say what should happen and write the correct line. Finally check the answer is in simplest form, with no square factor under a root and no surd in the denominator.
Method: 1. Replace each surd with a decimal estimate in the original and in your answer. 2. Compare the two decimals and look for a mismatch. 3. For a mistake question, find the first line that goes wrong and correct it. 4. Check for square factors under roots and for roots in the denominator.
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