POP

C1.7 Indices I

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01 — Explanation

What do positive, zero and negative indices mean?

An index tells you how many times a base is multiplied by itself: $a^n$ means $a$ multiplied by itself $n$ times.

A zero index always gives $1$ : $a^0 = 1$ (for $a \neq 0$ ), e.g. $9^0 = 1$ .
A negative index means "take the reciprocal of the positive power": $a^{-n} = \dfrac{1}{a^n}$ , e.g. $2^{-3} = \dfrac{1}{8}$ .

Core tier only covers positive, zero and negative integer indices — fractional indices are Extended-only.

Common Slip

the word "negative" makes people think the answer must be negative. It doesn't — $4^{-2} = \frac{1}{16}$ , which is positive. A negative index only ever flips the number to its reciprocal.

02 — Explanation

What are the index laws for multiplying and dividing powers?

These only work when the base is the same:

a^m \times a^n = a^{m+n} \qquad \text{e.g.} \quad x^2 \times x^5 = x^7

a^m \div a^n = a^{m-n} \qquad \text{e.g.} \quad y^8 \div y^3 = y^5

The base itself never changes — only the indices are added or subtracted. A common error is writing $3^2 \times 3^4 = 9^6$ ; the correct working keeps the base as $3$ : $3^2 \times 3^4 = 3^6$ .

03 — Explanation

How do I simplify a power raised to another power?

Multiply the indices together:

(a^m)^n = a^{mn} \qquad \text{e.g.} \quad (k^2)^4 = k^8

04 — Explanation

What happens when I try to combine powers using addition instead of multiplication?

The index laws for adding exponents only apply when the terms are being multiplied or divided — they do not apply to addition or subtraction of powers. This is one of the most common sources of lost marks in this topic.

2^3 + 2^2 \neq 2^5

To check: $2^3+2^2 = 8+4=12$ , but $2^5=32$ — clearly not equal. When terms are added, you must work out each power's actual value first and then add the results; there is no shortcut rule for combining the indices themselves.

Common Slip

applying $a^m \times a^n = a^{m+n}$ to a sum like $a^m + a^n$ . The index laws are strictly for multiplication and division of powers with the same base — always check which operation is actually being used before reaching for a law.

05 — Explanation

How do I combine more than one index law in the same question?

Work through the expression one operation at a time, applying the relevant law at each stage rather than trying to do everything in one jump.

Example: simplify $\dfrac{a^5 \times a^2}{a^4}$ .

Apply the multiplication law to the numerator first: $a^5 \times a^2 = a^{5+2}=a^7$ .
Now apply the division law: $\dfrac{a^7}{a^4}=a^{7-4}=a^3$ .

06 — Worked example

Worked exam-style question

(a) Find the value of $2^{-3}$ and $5^0 \times 3^2$ .

(b) Simplify $(2^3)^2$ and $\dfrac{x^6}{x^9}$ .

(a) $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$ . Also, $5^0 = 1$ , so $5^0 \times 3^2 = 1 \times 9 = 9$ .

(b) $(2^3)^2 = 2^{3\times2} = 2^6 = 64$ . Using the division law, $\dfrac{x^6}{x^9} = x^{6-9} = x^{-3} = \dfrac{1}{x^3}$ .

07 — Common mistakes

Common mistakes to avoid

Multiplying the base itself when multiplying two powers, e.g. writing $3^2 \times 3^4 = 9^6$ instead of $3^6$ .
Believing a negative index makes the result negative — it means reciprocal, not negative.
Believing $a^0 = 0$ . Any non-zero number to the power $0$ equals $1$ , which follows from $\frac{a^m}{a^m} = a^0 = 1$ .