# Multiplication using Exponents (Indices)

If we wish to calculate $5^4 \times 5^3$, we could write in factor form to get:
\begin{align} \displaystyle 5^4 \times 5^3 &= (5 \times 5 \times 5 \times 5) \times (5 \times 5 \times 5) \\ &= 5^7 \end{align}

### Example 1

Simplify $7^2 \times 7^3$ after first writing in factor form.

\begin{align} \displaystyle 7^2 \times 7^3 &= (7 \times 7) \times (7 \times 7 \times 7) \\ &= 7^5 \end{align}

However, if we look closely, a much simpler method would be to add the exponents since the bases are the same. Therefore this calculation can also be done this way:
\begin{align} \displaystyle 5^4 \times 5^3 &= 5^{4+3} \\ &= 5^7 \end{align}
, which is the same answer.

We can add the exponents when multiplying only if the bases are the same. Thus to $\textit{multiply}$ numbers with the $\textit{same base}$, keep the base and $\textit{add}$ the exponents.
$$\large a^x \times a^y = a^{x+y}$$

### Example 2

Simplify $9^3 \times 9^5$.

\begin{align} \displaystyle 9^3 \times 9^5 &= 9^{3+5} \\ &= 9^8 \end{align}

### Example 3

Simplify $4^3 \times 4 \times 4^5$.

\begin{align} \displaystyle 4^3 \times 4 \times 4^5 &= 4^3 \times 4^1 \times 4^5 \\ &= 4^{3+1+5} \\ &= 4^9 \end{align}

Many questions will be algebraic, meaning that a pronumeral is used. In such questions, we multiply the coefficients and apply the multiplication rule to the pronumeral separately.

### Example 4

Simplify $3x^5 \times 5x^4$.

\begin{align} \displaystyle 3x^5 \times 5x^4 &= (3 \times 5) \times (x^5 \times x^4) \\ &= 15 \times x^{5+4} \\ &= 15x^9 \end{align}

When more than one pronumeral is involved in the question, we apply this rule to each pronumeral separately.

### Example 5

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

\begin{align} \displaystyle 3x^2 \times 2x^5 \times x \times x^3 &= (3 \times 2) \times (x^2 \times x^5 \times x^1 \times x^3) \\ &= 6 \times x^{2 + 5 + 1 +3} \\ &= 6x^{11} \end{align}

### Example 6

Expand $x^2(x^3 + 4x^5)$.

\begin{align} \displaystyle x^2(x^3 + 4x^5) &= x^2 \times x^3 + x^2 \times 4x^5 \\ &= x^{2+3} + 4x^{2+5} \\ &= x^5 + 4x^7 \end{align}