Using the Quotient Rule of Exponents

The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as [latex]\frac{{y}^{m}}{{y}^{n}}[/latex], where [latex]m>n[/latex]. Consider the example [latex]\frac{{y}^{9}}{{y}^{5}}[/latex]. Perform the division by canceling common factors.

[latex]\begin{array}\text{ }\frac{y^{9}}{y^{5}}\hfill&=\frac{y\cdot y\cdot y\cdot y\cdot y\cdot y\cdot y}{y\cdot y\cdot y\cdot y\cdot y} \\ \hfill&=\frac{\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot y\cdot y\cdot y\cdot y}{\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}\cdot\cancel{y}} \\ \hfill& =\frac{y\cdot y\cdot y\cdot y}{1} \\ \hfill& =y^{4}\end{array}[/latex]

Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.

[latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]

In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.

[latex]\frac{{y}^{9}}{{y}^{5}}={y}^{9 - 5}={y}^{4}[/latex]

For the time being, we must be aware of the condition [latex]m>n[/latex]. Otherwise, the difference [latex]m-n[/latex] could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.

A General Note: The Quotient Rule of Exponents

For any real number [latex]a[/latex] and natural numbers [latex]m[/latex] and [latex]n[/latex], such that [latex]m>n[/latex], the quotient rule of exponents states that

[latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]

Example 2: Using the Quotient Rule

Write each of the following products with a single base. Do not simplify further.

  1. [latex]\frac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}[/latex]
  2. [latex]\frac{{t}^{23}}{{t}^{15}}[/latex]
  3. [latex]\frac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}[/latex]

Solution

Use the quotient rule to simplify each expression.

  1. [latex]\frac{{\left(-2\right)}^{14}}{{\left(-2\right)}^{9}}={\left(-2\right)}^{14 - 9}={\left(-2\right)}^{5}[/latex]
  2. [latex]\frac{{t}^{23}}{{t}^{15}}={t}^{23 - 15}={t}^{8}[/latex]
  3. [latex]\frac{{\left(z\sqrt{2}\right)}^{5}}{z\sqrt{2}}={\left(z\sqrt{2}\right)}^{5 - 1}={\left(z\sqrt{2}\right)}^{4}[/latex]

Try It 2

Write each of the following products with a single base. Do not simplify further.

  1. [latex]\frac{{s}^{75}}{{s}^{68}}[/latex]
  2. [latex]\frac{{\left(-3\right)}^{6}}{-3}[/latex]
  3. [latex]\frac{{\left(e{f}^{2}\right)}^{5}}{{\left(e{f}^{2}\right)}^{3}}[/latex]

Solution