Indices & Surds

What Your Child Is Learning

Indices are powers. When you see \(x^3\), it means \(x \times x \times x\). The small raised number is the index (plural: indices). It tells you how many times to multiply the base by itself.

HigherSurds are roots that can't be written as exact decimals. Instead of writing 1.41421356... your child writes \(\sqrt{2}\). Leaving the answer as a surd keeps it exact — no rounding, no loss of precision. This matters in GCSE maths because exam questions often say "give your answer in exact form".

How Schools Teach It Now

Students learn six index laws that cover most of what they will meet:

• Multiply: add the powers — \(x^a \times x^b = x^{a+b}\)
• Divide: subtract the powers — \(x^a \div x^b = x^{a-b}\)
• Power of a power: multiply the powers — \((x^a)^b = x^{ab}\)
• Negative index: take the reciprocal — \(x^{-n} = \dfrac{1}{x^n}\)
• HigherFractional index: means a root — \(x^{\frac{1}{n}} = \sqrt[n]{x}\)
• Zero index: anything except zero, to the power 0, is 1 — \(x^0 = 1\) for \(x \neq 0\)

HigherFor surds, the key skill is simplifying by finding square factors:

Common Mistakes to Watch For

• Multiplying powers instead of adding: \(x^3 \times x^4 = x^{12}\) is wrong. The correct answer is \(x^{3+4} = x^7\). When you multiply terms, you add the indices.
• HigherThinking surds can't be simplified: students often leave \(\sqrt{8}\) as is, but \(8 = 4 \times 2\), so \(\sqrt{8} = 2\sqrt{2}\).
• Confusing negative indices with negative numbers: \(2^{-3}\) does not equal \(-8\). It equals \(\dfrac{1}{2^3} = \dfrac{1}{8}\). The negative index means "reciprocal", not "make it negative".

Try It Together

Grab a piece of paper and try this with your child:

Let Them Practise

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