Raising a Power to Another Power

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 using the multiplication using exponents 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
$$(2^3)^4 = 2^{3 \times 4} = 2^{12}$$ and $$(5^2)^3 = 5^{2 \times 3} = 5^6$$ When raising a power to another power, we multiply the exponents (indices). $$(a^x)^y = a^{x \times y}$$ This rule also implies; $$(a \times b)^x = a^x \times b^x$$ $$\Big(\dfrac{a}{b}\Big)^x = \dfrac{a^x}{b^x} $$

Example 1

Simplify $(6^3)^3$.

Example 2

Simplify $(ab^4)^3$.

Example 3

Simplify $(2a^3b^2)^3$.

Example 4

Simplify $(2x^3)^2 \times (3x^5)^3$.

Example 5

Simplify $\Big(\dfrac{2a^3}{b^2}\Big)^3$.