# 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
\large \begin{align} \require{AMSsymbols} \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

After first writing in factor form, simplify $\dfrac{x^7}{x^4}$.

\begin{align} \displaystyle \require{AMSsymbols} \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.

\large \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 the same bases.

\large \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 the multiplication of algebraic expressions, 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 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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