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Index law 3: A rule for raising a power to a power

Example 5

\begin{align}(8^3)^4 &= 8^3 × 8^3 × 8^3 × 8^3\\ &= 8^{3+3+3+3}\hspace{20mm} \text{(using index law 1)}\\ &= 8^{3×4}\\ &= 8^{12}\end{align}

Index law 3

When a power is raised to another power, multiply the indices.

\((a^m)^n = a^{m×n}\)

 

Example 6

Write \((5^3)^2\) as a single power of 5.

Solution

\begin{align}(5^3)^2 &= 5^{3×2}\\ &= 5^6\end{align}

Index law 4: Powers of products

It is often useful to expand a product.

Example 7

\begin{align}(3 × 4)^5 &= (3 × 4) × (3 × 4) × (3 × 4) × (3 × 4) × (3 × 4)\\ &= (3 × 3 × 3 × 3 × 3) × (4 × 4 × 4 × 4 × 4)\\ &= 3^5 × 4^5\end{align}

 

Index law 4

A power of a product is the product of the powers.

\((a × b)^n = a^n × b^n\)

Example 8

Use index law 4 to evaluate \(5^4 × 2^4.\)

Solution

\begin{align}5^4 × 2^4 &= (5 × 2)^4\\ &= 10^4\\ &= 10 000\end{align}

Index law 5: Powers of quotients

The brackets in a power of a quotient can also be expanded.

Example 9

\begin{align}(\dfrac{2}{3})^5 &=\dfrac{2}{3} × \dfrac{2}{3} × \dfrac{2}{3} × \dfrac{2}{3} × \dfrac{2}{3}\\ &=\dfrac{2 × 2 × 2 × 2 × 2}{3 × 3 × 3 × 3 × 3}\\ &=\dfrac{2^5}{3^5}\end{align}

Index law 5

A power of a quotient is the quotient of the powers.

\((\dfrac{a}{b})^n = \dfrac{a^n}{b^n}\)

Example 10

Evaluate \(\dfrac{20^3}{5^3}\)

Solution

\begin{align}\dfrac{20^3}{5^3} &= (\dfrac{20}{5})^3\\ &= 4^3\\ &= 64\end{align}