E1.7Extended tier0580 · Section 1: Number

Indices I — positive, zero, negative and fractional indices

Key learnings

Solve index equations by matching the base
Write both sides as powers of the same base, then equate the exponents (e.g. $9^x = 27 \Rightarrow 3^{2x} = 3^3 \Rightarrow 2x = 3$ ).
Negative and fractional indices
$a^{-n} = \dfrac{1}{a^n}$ (a reciprocal, not a negative number); $a^{1/n} = \sqrt[n]{a}$ ; and $a^{m/n} = \left(\sqrt[n]{a}\right)^m$ .
The laws of indices
$a^m \times a^n = a^{m+n}$ , $\ a^m \div a^n = a^{m-n}$ , $\ (a^m)^n = a^{mn}$ , and $a^0 = 1$ .

Common mistakes

Drawn from real examiner reports.

Forgetting root-then-power order

For $a^{m/n}$ take the root first, then the power: $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$ . Doing the power first ( $16^3$ then the 4th root) gives huge numbers and invites arithmetic slips.

Not rewriting numbers to a common base

Failing to express numbers like $9$ , $27$ or $81$ as powers of $3$ , so the equation can't be solved by matching exponents.

A commonly reported examiner note records that powers of 9, 27 and 81 left un-converted blocked progress.

Negative index treated as negative

Thinking $a^{-2}$ is negative rather than $\tfrac{1}{a^2}$ ; or forgetting the root in a fractional index.

Multiplying powers instead of adding

When multiplying like bases the indices ADD; students often multiply them (or vice versa), e.g. mishandling $p^a \times p^b$ .

A commonly reported examiner note records this, with powers were "incorrectly multiplied instead of added".

Not flipping a fraction for a negative index

A negative index on a fraction flips it: $\left(\tfrac{8}{27}\right)^{-2/3} = \left(\tfrac{27}{8}\right)^{2/3}$ . Leaving the fraction the same way up and just making the answer negative is wrong — the index is a reciprocal instruction, not a sign.

Exam tips

Rewrite to a common base first
Before comparing or simplifying, turn every term into a power of the same base — then the index laws do the work.
Split coefficient and letter separately
In $\tfrac{12x^5}{3x^2}$ , handle the number and the letter apart: divide $12 \div 3 = 4$ and subtract indices $x^{5-2} = x^3$ , giving $4x^3$ . Applying the index law to the coefficient too is a common slip.
Zero and negative indices: sanity-check
Remember $a^0 = 1$ for any non-zero $a$ , and that a negative index gives a fraction below $1$ , not a negative value ( $5^{-2} = \tfrac{1}{25}$ ). A quick sanity check catches sign and reciprocal errors.

Revision notes

For the same base $a$ : $a^m \times a^n = a^{m+n}, \qquad a^m \div a^n = a^{m-n}, \qquad (a^m)^n = a^{mn}, \qquad a^0 = 1.$ Multiply → add the powers. Divide → subtract. Power of a power → multiply. Mixing these up (adding when you should multiply, or the reverse) is the most penalised slip.

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Indices I — positive, zero, negative and fractional indices

Key learnings

Common mistakes

Exam tips

Revision notes

The laws of indices

Negative and fractional indices

Solving index equations

Flashcards

Practice questions