If we are given $(2^3)^4$, that can be written in factor form as $2^3 \times 2^3 \times 2^3 \times 2^3$.
We can then simplify the multiplication using the exponent’s rule as $2^{3+3+3+3} = 2^{12}$.
Similarly, if we are given $(5^2)^3$, this means;
\( \begin{align}
(5^2)^3 &= 5^2 \times 5^2 \times 5^2 \\
&= 5^{2+2+2} \\
&= 5^6
\end{align} \)
Using the above method we can see that $(2^3)^4 = 2^{12}$ and $(5^2)^3 = 5^6$.
You will notice that
$$ \large \begin{align} (2^3)^4 &= 2^{3 \times 4} = 2^{12} \\
(5^2)^3 &= 5^{2 \times 3} = 5^6 \end{align} $$
When raising a power to another power, we multiply the exponents (indices).
$$ \large (a^x)^y = a^{x \times y}$$
This rule also implies;
$$ \large \begin{align} (a \times b)^x &= a^x \times b^x \\
\Big(\dfrac{a}{b}\Big)^x &= \dfrac{a^x}{b^x} \end{align} $$
Example 1
Simplify $(6^3)^3$.
\( \begin{align} \displaystyle
(6^3)^3 &= 6^{3 \times 3} \\
&= 6^9
\end{align} \)
Example 2
Simplify $(ab^4)^3$.
\( \begin{align} \displaystyle
(ab^4)^3 &= (a^1b^4)^3 \\
&= a^{1 \times 3} b^{4 \times 3} \\
&= a^3b^{12}
\end{align} \)
Example 3
Simplify $(2a^3b^2)^3$.
\( \begin{align} \displaystyle
(2a^3b^2)^3 &= 2^3 a^{3 \times 3} b^{2 \times 3} \\
&= 8a^9b^{6}
\end{align} \)
Example 4
Simplify $(2x^3)^2 \times (3x^5)^3$.
\( \begin{align} \displaystyle
(2x^3)^2 \times (3x^5)^3 &= 2^2 x^{3 \times 2} \times 3^3 x^{5 \times 3} \\
&= 4x^6 \times 27x^{15} \\
&= (4 \times 27) \times x^{6+15} \\
&= 108x^{21}
\end{align} \)
Example 5
Simplify $\Big(\dfrac{2a^3}{b^2}\Big)^3$.
\( \begin{align} \displaystyle
\Big(\dfrac{2a^3}{b^2}\Big)^3 &= \dfrac{(2a^3)^3}{(b^2)^3} \\
&= \dfrac{2^3 a^{3 \times 3}}{b^{2 \times 3}} \\
&= \dfrac{8 a^9}{b^6}\
\end{align} \)
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