Indices

Powers, the index laws for products, quotients and powers, and the meaning of zero and negative indices.

Power, base and index

In $2^5$, the base is $2$, the index (or exponent) is $5$, and the whole expression is a power: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$.

Index laws

$$a^m \times a^n = a^{m+n} \qquad a^m \div a^n = a^{m-n} \qquad (a^m)^n = a^{mn}$$ $$(ab)^n = a^n b^n \qquad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Zero and negative indices

Each step left halves the value

Move P along y = 2ˣ. Each step to the left divides by 2, which is why 2⁰ = 1 and 2⁻¹ = ½.

Try this

Find $2^{-3}$ on the graph before you calculate it.

Example 1●

Simplify $\dfrac{(2x^3)^4}{8x^5}$.

$(2x^3)^4 = 16x^{12}$

$\dfrac{16x^{12}}{8x^5} = 2x^7$

Example 2●●

Evaluate $\left(\dfrac{2}{3}\right)^{-2}$ and write $\dfrac{5}{x^3}$ with a negative index.

$\left(\dfrac{2}{3}\right)^{-2} = \left(\dfrac{3}{2}\right)^2 = \dfrac{9}{4}$

$\dfrac{5}{x^3} = 5x^{-3}$

Flashcards

$a^m imes a^n$

$a^{m+n}$

$a^m div a^n$

$a^{m-n}$

$(a^m)^n$

$a^{mn}$

$a^0$

$1$ (for $a \ne 0$)

$a^{-n}$

$\dfrac{1}{a^n}$

Practice

Simplify $a^5 \times a^3$.
Simplify $(2x^3)^4$.
Evaluate $5^0$.
Evaluate $2^{-3}$ as a fraction.
Evaluate $\left(\tfrac23\right)^{-2}$.
Simplify $x^7 \div x^{-2}$.
Write $\dfrac{1}{x^4}$ as a power of $x$.
Simplify $(3a^2 b)^2 \times a^{-1}$.
Evaluate $\dfrac{10^4 \times 10^{-6}}{10^{-3}}$.