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.
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. $$a^x \times a^y = a^{x+y}$$

Example 2

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

Example 3

Simplify $4^3 \times 4 \times 4^5$.
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 $3x^5 \times 5x^4$.
When there is more than one pronumeral involved in the question, we apply this rule to each pronumeral separately.

Example 5

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

Example 6

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