Identity

Power of a power, and the move to a common base

Algebra • Indices

Formula

(a^{m})^{n} = a^{mn}

For

a > 0

and any

m

and

n

(a^{m})^{n} = a^{mn}

; so

4^{k} = 2^{2 k}

8^{k} = 2^{3 k}

2 7^{k} = 3^{3 k}

, and powers written to different bases can be brought to one base and then compared or combined.

How to use it

1Rewrite every base as a power of the smallest base present, then collect the indices.
2Once the bases agree, $a^{p} = a^{q}$ forces $p = q$ for $a > 0$ and $a \neq = 1$ .
3To handle a sum of powers, factor out the lowest one: $2^{10} + 2^{12} = 2^{10} (1 + 2^{2})$ .

Write

\frac{8 ^{7}}{4 ^{9}}

as a power of

2

Step-by-step

1
$8^{7} = (2^{3})^{7} = 2^{21}$ and $4^{9} = (2^{2})^{9} = 2^{18}$
Why: Multiply the indices to put both terms on the base $2$ .
2
$\frac{2 ^{21}}{2 ^{18}} = 2^{3}$

Answer

2^{3}

Watch out

(a^{m})^{n} = a^{mn}

, but

a^{m^{n}}

is something else:

(2^{3})^{2} = 64

while

2^{3^{2}} = 512

. And

a^{m} \times a^{n} = a^{m + n}

adds the indices; only a power of a power multiplies them.

Also known as

index laws, laws of indices, power of a power, common base, express as a power of 2