The Power of a Power Rule

A power, raised to a power

${\left(2^3\right)}^2$(2³)² means $2^3$2³ squared: two copies of $2^3$2³ multiplied together, $2^3\times 2^3$2³ × 2³. The brackets make the whole of $2^3$2³ the base, and the index 2 outside the brackets counts copies of it.

Two copies of the whole of $2^3$2³, multiplied together.

Unpack the copies

Each $2^3$2³ is three 2s multiplied together. Two groups of three 2s make 3 × 2 = 6 copies of 2, so ${\left(2^3\right)}^2=2^6$(2³)² = 2⁶.

Check with the values: $2^3=8$2³ = 8, and $8^2=8\times 8=64$8² = 8 × 8 = 64, which is $2^6$2⁶.

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

The rule

The same happens for any base and any indices. ${\left(a^m\right)}^n$(aᵐ)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: ${\left(a^m\right)}^n=a^{mn}$(aᵐ)ⁿ = aᵐⁿ. It is the one index law that multiplies the indices.

So ${\left(x^3\right)}^4=x^{12}$(x³)⁴ = x¹², four groups of three copies, and ${\left(5^2\right)}^3=5^6$(5²)³ = 5⁶, 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$2³ × 2² puts 3 copies beside 2 copies in one row: 3 + 2 = 5, so it is $2^5$2⁵. ${\left(2^3\right)}^2$(2³)² makes 2 groups of 3 copies: 3 × 2 = 6, so it is $2^6$2⁶.

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$2³ × 2² = 2⁵.

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. ${\left(5^2\right)}^3$(5²)³ is not $5^5$5⁵. $5^5$5⁵ is $5^2\times 5^3$5² × 5³, two powers side by side, not three groups of $5^2$5².

Raising one index to the other. ${\left(5^2\right)}^3$(5²)³ is not $5^8$5⁸, which comes from $2^3=8$2³ = 8. The indices multiply as ordinary numbers, 2 × 3 = 6, so ${\left(5^2\right)}^3=5^6$(5²)³ = 5⁶.