GCSE · Number

Surds Higher

Some square roots, like √9 = 3, come out as neat numbers. Most, like √2, never do: their decimals go on for ever without a pattern. Rather than round them, mathematicians keep them as roots and work with them exactly. These exact roots are called surds, and this page shows you how to simplify them, add and multiply them, and tidy them out of the bottom of a fraction.

DfE GCSE subject content (Number, Higher tier): calculate exactly with surds; simplify surd expressions involving squares, e.g. √12 = √(4 × 3) = √4 × √3 = 2√3; rationalise denominators

What you need from KS3

1 · What is a surd? Higher

Draw a square with sides 1 unit long and measure across its diagonal. By Pythagoras (see the Pythagoras page), the diagonal d satisfies d² = 1² + 1² = 2, so d = √2. That is a real, physical length. Swing the diagonal down onto the number line and see exactly where it lands.

So what is √2 as a decimal? We can trap it. 1.4² = 1.96 is too small and 1.5² = 2.25 is too big, so √2 is between 1.4 and 1.5. Press the button to zoom in one decimal place at a time.

However far you zoom, the square is never exactly 2. The decimal for √2 goes on for ever and never settles into a repeating pattern. A number like that cannot be written as a fraction, and is called irrational. (Numbers that can be written as a fraction of two whole numbers, like 3, ½ or 0.75 = ¾, are rational.)

Why √2 can't be a fraction. Suppose √2 = p/q, a fraction already in its lowest terms. Square both sides: 2 = p²/q², so p² = 2q². That makes p² even, so p is even: write p = 2k. Then 4k² = 2q², so q² = 2k², which makes q even too. But then p and q share a factor of 2, and we said the fraction was in lowest terms. The assumption has led to nonsense, so it was false: √2 is not a fraction.

A surd is a root that is irrational, such as √2, √3, √10 or 5√7. √9 is not a surd, because it equals 3 exactly. The square root of any whole number that isn't a square number is a surd. Slide n to see which roots are surds and where they sit on the number line.

Why keep surds at all? Because any decimal for √2 is rounded, and rounding errors grow when you keep calculating. Writing √2 is exact, in the same way that leaving an answer as 12π is exact. Exam questions that say "give your answer in surd form" or "exact value" want you to keep the root.

2 · The key rule: √(ab) = √a × √b Higher

Every technique on this page rests on one rule: the square root of a product is the product of the square roots.

√(ab) = √a × √b    and    √(a ÷ b) = √a ÷ √b

Why it's true. Square the right-hand side: (√a × √b)² = √a × √b × √a × √b = (√a × √a) × (√b × √b) = a × b. So √a × √b is a positive number whose square is ab. That is exactly what √(ab) means. Check with numbers: √4 × √9 = 2 × 3 = 6, and √36 = 6. ✓

Set the two numbers and compare the bars. Then switch to + to see the rule that does not exist.

Exam trap: √(a + b) is NOT √a + √b. √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5. Think of a right-angled triangle with sides 3 and 4: going round the two short sides (7) is longer than cutting across the hypotenuse (5). Roots split over × and ÷, never over + or −.

3 · Simplifying surds Higher

A surd is in its simplest form when the number under the root has no square factor except 1. To simplify, find the largest square number that divides into the number, split the root using the key rule, and square-root the square part.

√72 = √(36 × 2) = √36 × √2 = 6√2

Here is a picture of why this works. 72 = 36 × 2, so you can build an area of 72 from 36 small squares, each of area 2, arranged 6 by 6. Each small square has side √2, so the big square has side 6 × √2. Its area is 72, so its side is also √72. Both are the same length: √72 = 6√2.

Tap a smaller square factor (for 72, try 4) and you'll see it still works, but you're left with √18, which can be simplified again. Using the largest square factor does it in one go. If you can't spot it, use prime factors: 72 = 2 × 2 × 2 × 3 × 3. Each pair of equal primes comes out of the root as one number: a pair of 2s gives 2, a pair of 3s gives 3, and the lonely 2 stays in. 2 × 3 × √2 = 6√2.

Common mistakes when simplifying

4 · Adding and subtracting surds Higher

Surds behave like algebra. Just as 3x + 5x = 8x, you can add like surds, ones with the same number under the root: 3√2 + 5√2 = 8√2. Unlike surds such as √2 and √3 can't be combined, just as 3x + 5y stays as it is. Often surds only look unlike until you simplify them. Step through each example; every orange rod has the length of one root.

Exam trap: √2 + √3 is not √5, and √8 − √2 is not √6. You can only add the numbers in front of identical roots. Always simplify every surd first; that is how √12 + √27 turns into 2√3 + 3√3 = 5√3.

5 · Multiplying surds and expanding brackets Higher

To multiply, multiply the numbers outside together and the numbers inside together (the key rule again), then simplify:

2√3 × 4√5 = 8√15

One special case is worth learning by heart: a surd times itself is the number inside, √3 × √3 = 3, because that's what a square root means. So brackets with surds expand exactly like double brackets in algebra (FOIL or the grid method), and any √n × √n term turns into an ordinary number. Change the numbers or tap a preset.

Common mistakes with brackets

6 · Rationalising the denominator: a√b Higher

A fraction like 6√2 has a surd in its denominator (the bottom). It is perfectly correct, but convention says to rewrite it with a whole number on the bottom. This is called rationalising the denominator. It makes answers easier to compare and to add, and before calculators it turned "divide by 1.41421…" into "divide by 2".

The trick: multiply by √b over √b. That fraction equals 1, so the value doesn't change, but the bottom becomes √b × √b = b, a whole number.

Common mistakes: multiplying only the top by √b (that changes the value, so you must do both); writing √b × √b = √(b²) and stopping (finish it: it's b); and forgetting to cancel at the end, e.g. 6√2/2 should become 3√2.

7 · Rationalising ab ± √c with the conjugate Higher

Now the bottom is something like 3 − √5. Multiplying by √5 won't help: (3 − √5) × √5 = 3√5 − 5, still a surd. Instead, use the result from section 5. Multiply by the conjugate, the same bracket with the sign in the middle swapped: the conjugate of 3 − √5 is 3 + √5. Then the middle terms cancel:

(3 − √5)(3 + √5) = 9 − 5 = 4

This is the difference of two squares, (x − y)(x + y) = x² − y², with y = √5 so y² is a whole number. As before, multiply top and bottom so the fraction's value doesn't change.

Exam traps: multiplying by the same bracket instead of the conjugate ((3 − √5)² = 14 − 6√5 still has a surd); forgetting to multiply out the numerator; and getting the sign of the new denominator wrong when c is bigger than b² (for example 2 − √7 gives 4 − 7 = −3, a negative denominator).

8 · Practice Higher

New questions each time. Type your answer as (number) √ (number), and for rationalising questions also fill in the denominator (leave it as 1 if it cancels completely). The feedback tries to name your mistake.

Worked examples

Simplify √72 fully Higher
List the square numbers that divide 72: 4 (72 = 4 × 18), 9 (72 = 9 × 8), 36 (72 = 36 × 2). The largest is 36.
Split the root: √72 = √(36 × 2) = √36 × √2.
Root the square: √36 = 6, so √72 = 6√2.
Check: 6² × 2 = 36 × 2 = 72 ✓, and 2 has no square factor, so this is simplest. (If you'd used 4 first: √72 = 2√18 = 2 × √9 × √2 = 2 × 3√2 = 6√2. Same answer, one extra step.)
Simplify √75 + √48 − √27 Higher
Simplify each surd: √75 = √(25 × 3) = 5√3; √48 = √(16 × 3) = 4√3; √27 = √(9 × 3) = 3√3.
They are all like surds (√3), so collect: 5√3 + 4√3 − 3√3 = (5 + 4 − 3)√3.
Answer: 6√3.
Expand and simplify (3 + 2√5)(1 − √5) Higher
Four products: 3 × 1 = 3; 3 × (−√5) = −3√5; 2√5 × 1 = 2√5; 2√5 × (−√5) = −2 × 5 = −10.
Collect whole numbers: 3 − 10 = −7. Collect surds: −3√5 + 2√5 = −√5.
Answer: −7 − √5.
Check with a calculator: (3 + 4.472…)(1 − 2.236…) = −9.236…, and −7 − 2.236… = −9.236… ✓
Show that (√12 + √3)² = 27 Higher
Simplify inside first: √12 = √(4 × 3) = 2√3, so √12 + √3 = 2√3 + √3 = 3√3.
Square: (3√3)² = 3² × (√3)² = 9 × 3 = 27 ✓
Why this order? Expanding (√12 + √3)(√12 + √3) works too (12 + √36 + √36 + 3 = 12 + 6 + 6 + 3 = 27), but simplifying first is quicker. In a "show that" question, write every line.
Rationalise the denominator of 32√6 Higher
Multiply top and bottom by √6 (just the surd part, not the 2): 32√6 × √6√6 = 3√62 × 6 = 3√612.
Cancel the common factor 3: 3√612 = √64.
Check: 3 ÷ (2 × 2.449…) = 0.6123…, and 2.449… ÷ 4 = 0.6123… ✓
Rationalise 43 − √5, giving your answer in its simplest form Higher
Conjugate of 3 − √5 is 3 + √5. Multiply top and bottom by it.
Top: 4(3 + √5) = 12 + 4√5.
Bottom: (3 − √5)(3 + √5) = 9 + 3√5 − 3√5 − 5 = 4.
Divide: 12 + 4√54 = 3 + √5 (divide both terms on top by 4).
Check: 4 ÷ (3 − 2.236…) = 5.236…, and 3 + 2.236… = 5.236… ✓
Exam-style: write 2 + √32 − √3 in the form a + b√3 Higher
Multiply top and bottom by the conjugate 2 + √3.
Top: (2 + √3)² = 4 + 2√3 + 2√3 + 3 = 7 + 4√3.
Bottom: (2 − √3)(2 + √3) = 4 − 3 = 1.
Answer: 7 + 4√3, so a = 7, b = 4. Check: 3.732… ÷ 0.2679… = 13.928…, and 7 + 4 × 1.732… = 13.928… ✓

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