Fractional indices
The nth root of aᵐ is a^(m/n), so the laws of indices work for roots too. Pupils first move between the two forms, then work out values such as 8^(2/3) = 4 and 27^(−2/3) = 1/9.
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Teaching guide: surds and fractional indices
Surds are roots that cannot be written as whole numbers or fractions, such as √2 or ∛5. Pupils learn to write roots as powers with fractional indices, to simplify surds, to add like surds, to multiply and divide them and, finally, to rationalise the denominator, the core of higher-tier number work. This free surds worksheet generator makes A4 worksheets for every step, Grades 9 and 10, with an answer key that shows all the working and every answer in its simplest form.
- For
- Middle & High 9th, 10th (ages 14-15)
- Subject
- Mathematics › Algebra
- Includes
- Student sheet and answer key
- Format
- A4, to print or save as PDF; new exercises every time
- Price
- Free, no sign-up
- Available in
- Catalan, Spanish, English
Student worksheet 
Answer key Equivalent roots
Dividing the index and the power by the same number simplifies a root; multiplying them gives a common index, to compare roots or to multiply roots with different indices.
Simplifying surds
Write the number under the root as prime factors: each factor comes out as many times as the index fits into its power, √72 = √(2³ × 3²) = 6√2. Bringing a factor in is the way back.
Adding and subtracting
Only like surds can be added, so simplify each surd first: √12 + √27 = 2√3 + 3√3 = 5√3.
Multiplying and dividing
With the same index, multiply or divide the numbers under the roots; with different indices, use a common index first. Brackets with surds expand like algebraic brackets, with the special products.
Rationalising
To remove a root from the denominator, multiply by the same root, by the root that completes an exact power, or by the conjugate when the denominator is a sum or a difference.
Frequently asked questions about surds worksheets
Which grades are these worksheets for?
Grades 9 and 10, the higher tier of GCSE number work. Level 1 of simplifying surds, adding like surds and writing roots as powers suits Grade 9; a common index, rationalising, expanding brackets and negative fractional indices are for Grade 10.
Is every answer in its simplest form?
Yes. Every answer is worked out with exact numbers and given in its simplest form: every factor that can come out of the root is taken out, the index is as small as it can be and the denominator is rationalised. So there are never two different right answers to one question.
How do you rationalise the denominator?
For a square root, multiply the numerator and the denominator by the same root: 6/√3 = 6√3/3 = 2√3. For a root with another index, multiply by the root that makes the number under it an exact power: 5/∛2 = 5∛4/2. For a sum or a difference, multiply by the conjugate: the difference of two squares removes the root.
When can you add two surds?
When they are like surds: the same index and the same number under the root. √2 + √3 cannot be simplified, but √8 + √18 can, because simplifying each surd gives 2√2 + 3√2 = 5√2.
Why do the letters stand for positive numbers?
So that √(x²) = x with no absolute value. It is the usual convention when simplifying roots with letters, and level 3 of simplifying surds says so in the instruction.
Which other worksheets go well with these?
Before surds, the powers and square roots worksheets practise the laws of indices and standard form works with powers of 10. Alongside them, polynomials use the same special products and quadratic equations give answers as surds. You will find every resource in maths worksheets.
How do I print only the answer key?
Press Print and choose “Answer key”, or “Worksheet and answer key” for both. You can also save it as a PDF from the browser's print dialog. The worksheets are made in your browser and no data is sent.
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