Simplifying Expressions Using the Quotient Property of Exponents

So far we have developed the properties of exponents for multiplication. We summarize these properties here.

Now we will look at the exponent properties for division. We previously learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property.

As before, we’ll try to discover a property by looking at some examples.

Let’s look at dividing terms containing exponential expressions. What happens if we divide two numbers in exponential form with the same base? Consider the following expression.

[latex] \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}[/latex]

We can rewrite the expression as: [latex] \displaystyle \frac{4\cdot 4\cdot 4\cdot 4\cdot 4}{4\cdot 4}[/latex]. Then we can cancel the common factors of [latex]4[/latex] in the numerator and denominator: [latex] \displaystyle \frac{\color{red}{4}\cdot \color{red}{4}\cdot 4\cdot 4\cdot 4}{\color{red}{4}\cdot \color{red}{4}}[/latex]. This leaves [latex]4^{3}[/latex] on the numerator and [latex]1[/latex] on the denominator, which simplifies to [latex]4^{3}[/latex] using exponential notation. Notice that the exponent, [latex]3[/latex], is the difference between the two exponents in the original expression, [latex]5[/latex] and [latex]2[/latex].

[latex] \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[/latex].

As another example, consider [latex]\frac{{2}^{5}}{{2}^{2}}[/latex]:

[latex]\frac{{2}^{5}}{{2}^{2}}=\frac{\color{red}{2\cdot 2}\cdot 2\cdot 2\cdot 2}{\color{red}{2\cdot 2}}=\frac{2\cdot 2\cdot 2\cdot}{1}=2^3[/latex]

If we subtract the exponents and keep the common base we get [latex]\frac{{2}^{5}}{{2}^{2}}=2^{5-2}=2^3[/latex].  The same answer we got when we expanded the exponents into multiplication.

Notice that in each case the bases were the same and we subtracted the exponents.  So, to divide two exponential terms with the same base, subtract the exponents.

 

One as an Exponent

What does [latex]3^1[/latex] or [latex]{(-5)}^1[/latex] equal?

Consider [latex]\frac{2^4}{2^3}[/latex]. If we use the quotient property, [latex]\frac{2^4}{2^3}=2^1[/latex].

Alternatively, we could expand the exponential terms: [latex]\frac{2^4}{2^3}=\frac{2\cdot 2\cdot 2\cdot 2}{2\cdot 2\cdot 2}[/latex]. Then by cancelling out common factors, we get  [latex]\frac{\color{red}{2\cdot 2\cdot 2\cdot} 2}{\color{red}{2\cdot 2\cdot 2}}=2[/latex].

This means that [latex]2^1=2[/latex], and leads us to the property that, for any integer [latex]a\text{, }a^1=a[/latex].

Zero as an Exponent

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as  [latex]\frac{{5}^{4}}{{5}^{4}}[/latex]. From earlier work with fractions, we know that,

[latex]\frac{2}{2} =\frac{17}{17} =\frac{-43}{-43} =1[/latex]

In words, a non-zero integer divided by itself is [latex]1[/latex]. Remember that [latex]\frac{0}{0}[/latex] is undefined.

We also know that [latex]a\text{, }a^1=a[/latex], for any non-zero integer [latex]a[/latex].

Now consider simplifying the term [latex]\frac{8}{8}[/latex] in two different ways.

We know that [latex]\frac{8}{8}=1[/latex] by division.

We also know that [latex]\frac{8}{8}=\frac{8^1}{8^1}=8^{1-1}=8^0[/latex] using the Quotient Property..

This means that [latex]8^0=1[/latex].

Negative Exponents

Now let’s see what happens when the denominator has a larger exponent than the numerator, so that when we subtract the exponents, we get a negative integer:

[latex]\frac{{2}^{2}}{{2}^{3}}=\frac{\color{red}{2\cdot 2}}{\color{red}{2\cdot 2}\cdot 2}=\frac{1}{2}[/latex]

When we subtract the exponents and keep the common base we get:

[latex]\frac{{2}^{2}}{{2}^{3}}=2^{2-3}=2^{-1}[/latex]

This means that [latex]2^{-1}=\frac{1}{2}[/latex].

Let’s consider one more example:

[latex]\frac{{7}^{2}}{{7}^{5}}=\frac{\color{red}{7\cdot 7}}{\color{red}{7\cdot 7\cdot} 7\cdot 7\cdot 7}=\frac{1}{7^3}[/latex].

On the other hand, subtracting the exponents and keeping the common base gives us [latex]\frac{{7}^{2}}{{7}^{5}}=7^{2-5}=7^{-3}[/latex].

So, [latex]\frac{1}{7^3}=7^{-3}[/latex].

This leads us to the meaning of negative exponents.

When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, the negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. We rewrite it by using the definition of negative exponents, [latex]{a}^{-n}={\frac{1}{{a}^{n}}}[/latex].  Any expression that has negative exponents is not considered to be in simplest form, so we will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.

In the following video you will see examples of simplifying expressions with negative exponents.

We can now update the quotient property of exponents so that it includes negative and zero exponents by removing the condition that [latex]m\gt n[/latex].

Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

 

Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and [latex]1[/latex] in the numerator.

 

Simplifying Fractions Raised to a Power

Now we will look at an example that will lead us to the Quotient to a Power Property.

Let’s look at what happens if you raise a fraction to a power. Remember that a fraction bar means divide. Suppose we have [latex] \displaystyle \frac{5}{4}[/latex] and raise it to the power [latex]3[/latex].

[latex] \displaystyle {{\left( \frac{5}{4} \right)}^{3}}=\left( \frac{5}{4} \right)\left( \frac{5}{4} \right)\left( \frac{5}{4} \right)=\frac{5\cdot 5\cdot 5}{4\cdot 4\cdot 4}=\frac{{{5}^{3}}}{{{4}^{3}}}[/latex]

Raising the fraction to the power of [latex]3[/latex] can also be written as the numerator [latex]5[/latex] to the power of [latex]3[/latex], and the denominator [latex]4[/latex] to the power of [latex]3[/latex].

Notice that the exponent applies to both the numerator and the denominator.  This leads to the Quotient to a Power Property for Exponents.

Let’s looks at some examples of how this rule applies under different circumstances.