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

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

Example 5

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

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

Example 6

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

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

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