The Power of a Power Rule

A power, raised to a power

$\left(2^{3}\right)^{2}$ means $2^{3}$ squared: two copies of $2^{3}$ multiplied together, $2^{3} \times 2^{3}$. The brackets make the whole of $2^{3}$ the base, and the index 2 outside the brackets counts copies of it.

Two copies of the whole of $2^{3}$, multiplied together.

Unpack the copies

Each $2^{3}$ is three 2s multiplied together. Two groups of three 2s make 3 × 2 = 6 copies of 2, so $\left(2^{3}\right)^{2} = 2^{6}$.

Check with the values: $2^{3} = 8$, and $8^{2} = 8 \times 8 = 64$, which is $2^{6}$.

Each group holds three 2s, and two groups hold six.

The rule

The same happens for any base and any indices. $\left(a^{m}\right)^{n}$ is n copies of aᵐ, and each of those holds m copies of a. That is n groups of m, which is m × n copies of a in all: $\left(a^{m}\right)^{n} = a^{mn}$. It is the one index law that multiplies the indices.

So $\left(x^{3}\right)^{4} = x^{12}$, four groups of three copies, and $\left(5^{2}\right)^{3} = 5^{6}$, three groups of two copies.

Adding or multiplying the indices?

Two laws look alike, so compare them side by side. $2^{3} \times 2^{2}$ puts 3 copies beside 2 copies in one row: 3 + 2 = 5, so it is $2^{5}$. $\left(2^{3}\right)^{2}$ makes 2 groups of 3 copies: 3 × 2 = 6, so it is $2^{6}$.

Ask whether the copies sit side by side or come in groups. Side by side, add the indices. Groups of copies, multiply them.

Side by side: 3 copies and 2 copies make 5, so $2^{3} \times 2^{2} = 2^{5}$.

Drag m and n, and switch between the two laws. Side by side, m copies and n copies make m + n. In a power of a power, n rows of m copies make m × n.

The usual mistakes

Adding the indices. $\left(5^{2}\right)^{3}$ is not $5^{5}$. $5^{5}$ is $5^{2} \times 5^{3}$, two powers side by side, not three groups of $5^{2}$.

Raising one index to the other. $\left(5^{2}\right)^{3}$ is not $5^{8}$, which comes from $2^{3} = 8$. The indices multiply as ordinary numbers, 2 × 3 = 6, so $\left(5^{2}\right)^{3} = 5^{6}$.