Year 10A - Surds and fractional indices - Free practice

Year 10A Surds and Fractional Indices Worksheets & Practice

Master surds, irrational numbers and fractional indices, the foundation of senior Methods maths.

Quick answer

What are surds and fractional indices?

A surd is an irrational root that cannot be simplified to a whole number or fraction, such as √2 or ∛5. You simplify surds by factoring out perfect squares (e.g. √72 = √(36×2) = 6√2), combine like surds by adding coefficients, and write roots as fractional indices using x^(1/n) = ⁿ√x.

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Worked examples

Year 10A surds and fractional indices: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Simplifying and adding like surds
Simplify $\sqrt{50} + \sqrt{8}$ .
Show worked solution
Key idea: Break each surd into a perfect-square factor, simplify, then collect like surds.
1. Simplify $\sqrt{50}$ : $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$
2. Simplify $\sqrt{8}$ : $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$
3. Add like surds: $5\sqrt{2} + 2\sqrt{2} = 7\sqrt{2}$
Answer: $7\sqrt{2}$
Example 2 · Expanding brackets involving surds
Expand and simplify $(3 + \sqrt{5})(2 - 3\sqrt{5})$ .
Show worked solution
Key idea: Expand using FOIL (First, Outer, Inner, Last), then simplify using $\sqrt{5} \times \sqrt{5} = 5$ .
1. First terms: $3 \times 2 = 6$
2. Outer terms: $3 \times (-3\sqrt{5}) = -9\sqrt{5}$
3. Inner terms: $\sqrt{5} \times 2 = 2\sqrt{5}$
4. Last terms: $\sqrt{5} \times (-3\sqrt{5}) = -3 \times 5 = -15$
5. Collect like terms: $\begin{aligned}6 - 9\sqrt{5} + 2\sqrt{5} - 15 &= (6 - 15) + (-9 + 2)\sqrt{5} \\ &= -9 - 7\sqrt{5}\end{aligned}$
Answer: $-9 - 7\sqrt{5}$
Example 3 · Fractional indices in a real-life context
A storage container is a cube with a volume of $200$ cubic metres. The side length of a cube with volume $V$ is given by $s = V^{\frac{1}{3}}$ . Write the exact side length of this container in surd form, and then evaluate it correct to two decimal places.
Show worked solution
Key idea: $V^{\frac{1}{3}}$ means the cube root of $V$ , so $V^{\frac{1}{3}} = \sqrt[3]{V}$ .
1. Substitute $V = 200$ : $s = 200^{\frac{1}{3}} = \sqrt[3]{200}$
2. Write $200 = 8 \times 25$ , so: $\sqrt[3]{200} = \sqrt[3]{8 \times 25} = \sqrt[3]{8} \times \sqrt[3]{25} = 2\sqrt[3]{25}$
3. Evaluate numerically: $2\sqrt[3]{25} \approx 2 \times 2.9240 \approx 5.85 \text{ m}$
Answer: Exact side length is $2\sqrt[3]{25}$ m, which is approximately $5.85$ m.

Common mistakes

Where students usually lose marks

Adding surds without simplifying first

$\sqrt{50} + \sqrt{8} = \sqrt{58}$

You cannot add surds by adding the numbers under the root sign. Always simplify each surd fully first, then collect like surds. Here, $\sqrt{50} + \sqrt{8} = 5\sqrt{2} + 2\sqrt{2} = 7\sqrt{2}$ .

Forgetting that √a × √a = a, not √(a²) left unsimplified

$(\sqrt{5})^2 = \sqrt{25} = 5$ written as $\sqrt{25}$ and left there, or wrongly as $(\sqrt{5})^2 = 5^2 = 25$

When you square a surd, the result is simply the number under the root sign: $(\sqrt{5})^2 = 5$ . Do not square the number inside as well.

Confusing fractional index notation

$x^{\frac{1}{3}} = \frac{x}{3}$ or $x^{\frac{2}{3}} = \frac{x^2}{3}$

A fractional index is not a fraction of $x$ . The denominator of the index is the root and the numerator is the power: $x^{\frac{m}{n}} = (\sqrt[n]{x})^m$ . For example, $x^{\frac{2}{3}} = (\sqrt[3]{x})^2$ .

Incorrect expansion with a negative sign inside brackets

$(3 + \sqrt{5})(3 - \sqrt{5}) = 9 - 5\sqrt{5}$ (forgetting the difference of two squares pattern gives an integer)

Use the difference of two squares identity: $(a + b)(a - b) = a^2 - b^2$ . So $(3 + \sqrt{5})(3 - \sqrt{5}) = 9 - 5 = 4$ . The surd terms cancel, leaving a rational number.

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Your turn

Expand and simplify $(3 + \sqrt{5})(3 - \sqrt{5})$ .

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About this worksheet

Why we built it

Surds and fractional indices are part of the Year 10A extension strand, designed for students heading towards senior Methods or Specialist maths. These worksheets build from defining rational and irrational numbers through to confidently simplifying surds, performing operations with them, and converting between surd and fractional index notation. Every question comes with a full worked solution so students can pinpoint exactly where their reasoning went astray.

What's covered

Sub-skills your student will practise

✓Identifying rational and irrational numbers
✓Simplifying surds (e.g. √72 = 6√2)
✓Adding and subtracting like surds
✓Multiplying and dividing surds
✓Expanding brackets involving surds, including difference of two squares
✓Converting between surd form and fractional index form (e.g. x^(1/3) = ³√x)

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Frequently asked

Questions parents ask about surds and fractional indices

What is the difference between a rational and an irrational number?

A rational number can be written as a fraction of two integers, such as 3/4 or 2. An irrational number, like √2 or √5, cannot be expressed as an exact fraction and its decimal expansion goes on forever without repeating.

How do fractional indices connect to surds?

A fractional index is just another way to write a root. For example, x^(1/2) means √x, and x^(1/3) means ³√x. More generally, x^(m/n) means the nth root of x, raised to the power m. Switching fluently between both forms is a key skill for senior maths.

Why do students need to learn surds in Year 10A and not just use a calculator?

Leaving answers in surd form gives an exact value, which is essential in higher maths where rounding errors accumulate. Year 12 exams regularly require exact surd answers, and the algebraic reasoning practised here underpins work with functions and calculus.

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