Surds: Simplifying & Rationalising (Grade 11)

Simplifying surds using perfect-square factors, adding and subtracting like surds, multiplying them out, and rationalising a denominator — including the conjugate method.

A surd is a root that cannot be simplified to a rational number, like 2 or 18. Grade 11 asks you to write them in simplest form and, crucially, to remove them from a denominator.

1Simplifying a surd

ab = a × b · ab = ab

Look for the largest perfect square that divides into the number, then split the root.

Worked ExampleSimplify 72

1
Find the largest perfect square factor of 72: 36.
72 = 36 × 2.
2
72 = 36 × 2.
Split the root over the product.
3
= 62.
36 = 6 exactly.

⚠️

Using a smaller factor still works but takes longer: 72 = 4×18 = 218, which must then be simplified again. Always hunt the largest square first.

2Adding and subtracting

Surds behave like like terms: you can only add or subtract them once the surd part matches.

Worked ExampleSimplify 32 + 50

1
Simplify the second: 50 = 25×2 = 52.
Now both terms contain 2.
2
32 + 52 = 82.
Add the coefficients, keep the surd.

⚠️

a + b is not a + b. Check it: 9 + 16 = 3 + 4 = 7, but 25 = 5.

3Multiplying

Worked ExampleExpand (3 + 2)(3 − 5)

1
3×3 = 3.
A root times itself gives the number.
2
−53 + 23 = −33.
Collect the surd terms.
3
2 × (−5) = −10.
4
= 3 − 33 − 10 = −7 − 33.

4Rationalising the denominator

A final answer may not have a surd in the denominator. Multiply top and bottom by whatever clears it — this changes the form, not the value.

Worked ExampleRationalise 63

1
Multiply top and bottom by 3.
Multiplying by 33 = 1 leaves the value unchanged.
2
633.
3×3 = 3.
3
= 23.
Simplify the fraction.

5The conjugate (two terms in the denominator)

The conjugate of a + b is a − b
(a + b)(a − b) = a² − b

Worked ExampleRationalise 41 + 2

1
Multiply top and bottom by the conjugate 1 − 2.
A difference of squares kills the surd.
2
Denominator: (1)² − (2)² = 1 − 2 = −1.
3
Numerator: 4(1 − 2) = 4 − 42.
4
= 4 − 42−1 = −4 + 42.
Divide through by −1.