Division using Exponents (Indices)

If we are given $a^8 \div a^3$, we can also write this as $\dfrac{a^8}{a^3}$, which means $\dfrac{a \times a \times a \times a \times a \times a \times a \times a}{a \times a \times a}$.
As there are $8$ factors of $a$ on the top line (numerator), and $3$ factors of $a$ on the bottom line (denominator), we can cancel $3$ of them, giving us
$$ \begin{align} \displaystyle \require{cancel}
&\dfrac{\cancel{a} \times \cancel{a} \times \cancel{a} \times a \times a \times a \times a \times a}{\cancel{a} \times \cancel{a} \times \cancel{a}} \\
&= a \times a \times a \times a \times a \\
&= a^5 \\
\end{align} $$

Example 1

Simplify $\dfrac{x^7}{x^4}$ after first writing in factor form.

\( \begin{align} \displaystyle \require{cancel}
\dfrac{x^7}{x^4} &= \dfrac{x \times x \times x \times x \times x \times x \times x}{x \times x \times x \times x} \\
&= \dfrac{\cancel{x} \times \cancel{x} \times \cancel{x} \times \cancel{x} \times x \times x \times x}{\cancel{x} \times \cancel{x} \times \cancel{x} \times \cancel{x}} \\
&= x \times x \times x \\
&= x^3 \\
\end{align} \)

An alternative solution can be formed by examining the result. We can see that the exponent (index) in the answer is the result of subtracting the exponents (indices) in the question.

$$ \begin{align} \displaystyle \require{cancel}
a^8 \div a^3 &= a^{8-3} \\
&= a^5 \\
\end{align} $$

We can subtract the exponents (indices) when dividing bases that are the same.

$$ \begin{align} \displaystyle
a^x \div a^y &=\dfrac{a^x}{a^y} \\
&= a^{x-y} \\
\end{align} $$

Example 2

Simplify $8^7 \div 8^5$ using the law of exponents (index law).

\( \begin{align} \displaystyle \require{cancel}
8^7 \div 8^5 &= 8^{7-5} \\
&= 8^2 \\
\end{align} \)

Example 3

Simplify $\dfrac{10^9}{10^6}$ using the law of exponents (index law).

\( \begin{align} \displaystyle \require{cancel}
\dfrac{10^9}{10^6} &= 10^{9-6} \\
&= 10^3 \\
\end{align} \)

As with multiplication of algebraic expressions, when dividing we divide the coefficients normally before applying the law of exponent (index law). to each pronumeral separately.

Example 4

Simplify $24x^7 \div 8x^2$ using the law of exponents (index law).

\( \begin{align} \displaystyle \require{cancel}
24x^7 \div 8x^2 &= \dfrac{24x^7}{8x^2} \\
&= \dfrac{24}{8} \times \dfrac{x^7}{x^2} \\
&= 3x^{7-2} \\
&= 3x^5 \\
\end{align} \)

Example 5

Simplify $\dfrac{25x^6 \times 9y^{11}}{15x^4 \times 3y^5}$ using the law of exponents (index law).

\( \begin{align} \displaystyle \require{cancel}
\dfrac{25x^6 \times 9y^{11}}{15x^4 \times 3y^5} &= \dfrac{25 \times 9}{15 \times 3} \times \dfrac{x^6}{x^4} \times \dfrac{y^{11}}{y^5} \\
&= 5 \times x^{6-4} \times y^{11-5} \\
&= 5x^2y^6
\end{align} \)

In examples where the coefficients do not divide evenly, we simplify the fraction that is formed by them.

Example 6

Simplify $\dfrac{7x^3 \times 4x^6}{12x^5}$ using the law of exponents (index law).

\( \begin{align} \displaystyle \require{cancel}
\dfrac{7x^3 \times 4x^6}{12x^5} &= \dfrac{7 \times 4}{12} \times \dfrac{x^3 \times x^6}{x^5} \\
&= \dfrac{7}{3} \times x^{3 + 6 – 5} \\
&= \dfrac{7}{3}x^4 \\
\end{align} \)

Example 7

Evaluate $\dfrac{2^x}{2^{x-3}}$ using the law of exponents (index law).

\( \begin{align} \displaystyle \require{cancel}
\dfrac{2^x}{2^{x-3}} &= 2^{x-(x-3)} \\
&= 2^{x-x+3} \\
&= 2^3 \\
&= 8 \\
\end{align} \)

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