Higher Surds: Simplifying, Multiplying and Dividing
A surd is a root such as sqrt(2) or sqrt(20) that is irrational and cannot be written as an exact decimal or a simple fraction. Simplifying a surd means rewriting it as k*sqrt(n) where n has no square factors, while multiplying and dividing surds uses the rules sqrt(a) x sqrt(b) = sqrt(ab) and sqrt(a) / sqrt(b) = sqrt(a/b). It appears throughout GCSE Higher and AQA Further Maths papers.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Find the largest square factor of the number under the root; for example, for 50 the largest square factor is 25.
- Split the surd into two factors, one of which is the square factor, and take the square root of that factor to bring it outside the root sign.
- Check whether the surd left inside the root can be simplified again, and repeat the process if it can.
- To multiply two surds, multiply the numbers outside the root together and the numbers inside the root together, then simplify the result.
- To divide two surds, divide the numbers inside the root by each other (or divide the whole surds), then simplify the result fully.
- When expanding brackets containing surds, multiply out each term individually, then collect the rational terms and the surd terms separately.
Worked example
Simplify sqrt(48), then use your answer to work out sqrt(48) x sqrt(3), giving your answer as an integer.
- Find the largest square factor of 48: 48 = 16 x 3, and 16 is a perfect square.
- Write sqrt(48) = sqrt(16) x sqrt(3) = 4sqrt(3).
- Substitute this into the product: sqrt(48) x sqrt(3) = 4sqrt(3) x sqrt(3).
- Since sqrt(3) x sqrt(3) = 3, this becomes 4 x 3.
- Final answer: 12.
Practice questions
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Q1Simplify sqrt(32) fully.Show answer
Answer: 4sqrt(2) (32 = 16 x 2, and 16 is a square number)
Q2Simplify sqrt(147) fully.Show answer
Answer: 7sqrt(3) (147 = 49 x 3)
Q3Work out sqrt(2) x sqrt(18), giving your answer as an integer.Show answer
Answer: 6 (combines to sqrt(36))
Q4Work out sqrt(63) / sqrt(7), giving your answer as an integer.Show answer
Answer: 3 (combines to sqrt(9))
Q5Simplify fully 4sqrt(3) + 2sqrt(3) - sqrt(3).Show answer
Answer: 5sqrt(3) (collect the coefficients: 4 + 2 - 1)
Q6Expand and simplify (5 + sqrt(3))(5 - sqrt(3)).Show answer
Answer: 22 (difference of two squares: 25 - 3)
Exam-style questions
Written in the style of a GCSE Further Maths exam paper, with a full mark scheme.
Simplify sqrt(8) x sqrt(50), giving your answer as an integer.
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Given that k x sqrt(2) = sqrt(32) + sqrt(50), find the value of the integer k.
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A right-angled triangle has legs of length sqrt(72) cm and sqrt(8) cm. (a) Show that the hypotenuse of the triangle has length 4sqrt(5) cm. (b) Find the exact area of the triangle.
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See real GCSE Further Maths past-paper questions, with official mark schemes →
Free printable worksheet
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