Review question

# Can we simplify these expressions involving indices? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8779

## Solution

Given that $p=24$ and $q=-16$, simplify

1. $\dfrac{x^px^q}{x^7}$,

2. $(x^2)^p(x^3)^q$.

There are three identities we need for this question, namely $x^ax^b = x^{a+b},\qquad \frac{x^a}{x^b} = x^{a-b},\qquad \left( x^a \right)^b = x^{ab}.$

Applying them to $(a)$ we find

\begin{align*} \frac{x^px^q}{x^7}&=\frac{x^{p+q}}{x^7}\\ &=x^{p+q-7}\\ &=x^{1}=x. \end{align*}

For $(b)$ we have

\begin{align*} \left(x^2 \right)^p \left(x^3 \right)^q &= x^{2p}x^{3q}\\ &=x^{2p+3q} \\ &= x^0=1. \end{align*}