Lesson Objectives

- Demonstrate an understanding of Exponents
- Demonstrate an understanding of the Product Rule for Exponents
- Demonstrate an understanding of the Power Rules for Exponents
- Learn how to simplify an exponent of 0
- Learn how to simplify an expression with Negative Exponents
- Learn how to use the Quotient Rule for Exponents

## How to use the Quotient Rule for Exponents & Simplify with Negative Exponents

In our last lesson, we learned about the power rules and product rule for exponents. In this lesson, we will expand on our knowledge of the rules of exponents and learn about negative exponents, the power of zero, and the quotient rule for exponents.

3

What is the value of 3 to the power of (-4)? To understand negative exponents, let's think about a pattern:

3 • 3 • 3 • 3 = 3

3 • 3 • 3 = 3

3 • 3 = 3

3

Each time we reduce our exponent by 1, we divide by our base of 3. This is because we are removing a factor of 3 when we decrease the exponent by 1. When we go from 3

3 • 3 • 3 • 3 = 3

3 • 3 • 3 = 3

3 • 3 = 3

3

3

This works for any non-zero number. We can state that any non-zero number raised to the power of zero is 1. If we try to raise zero to the power of zero, we will have a problem. We can't divide 0 by 0, this is undefined. Therefore, we say zero raised to the power of zero is undefined. What happens if we continue and decrease the exponent by 1 to (-1)? We would continue the pattern. 1 would be divided by 3, and could be written as 1/3:

3 • 3 • 3 • 3 = 3

3 • 3 • 3 = 3

3 • 3 = 3

3

3

3

As we continue to decrease our exponent by 1, we continue the same process. Divide by the base (3) each time we reduce the exponent by 1:

3 • 3 • 3 • 3 = 3

3 • 3 • 3 = 3

3 • 3 = 3

3

3

3

3

3

3

Obviously, we will not be going through all this division each time we need to simplify with negative exponents. This was just to give you an understanding of where our simplified result comes from. When we want to simplify with negative exponents, we take the reciprocal of the base and make the exponent positive. Another way to think about this is by stating that we will drag the base and exponent across the fraction bar and make the exponent positive. If we wanted to simplify 3

Example 1: Simplify each. $$5^{-5}$$ Take the reciprocal of the base: $$5^{-5}\hspace{.25em}» \hspace{.25em}\frac{1}{5^{-5}}$$ Make the exponent positive: $$\frac{1}{5^{-5}}\hspace{.25em}» \hspace{.25em}\frac{1}{5^{5}}$$ $$5^{-5}=\frac{1}{5^5}$$ Alternatively, we can start by creating a fraction with 1 as the denominator. When we drag an exponential expression across a fraction bar, the base stays the same and we change the sign of the exponent: $$5^{-5}=\frac{5^{-5}}{1}$$ Now we can drag the 5

Example 3: Simplify each. $$\frac{1}{y^{-4}}$$ Bring the base y into the numerator, the (-4) becomes (+4): $$\frac{1}{y^{-4}}=\frac{y^4}{1}=y^4$$ Example 4: Simplify each. $$\frac{6x^{-5}}{y^{-2}}$$ In this case, we have two exponential expressions with negative exponents. We will move x into the denominator and make (-5) into (+5): $$\frac{6x^{-5}}{y^{-2}}\hspace{.25em}» \hspace{.25em}\frac{6}{x^5y^{-2}}$$ Notice how 6 in the numerator was not moved. The 6 is not raised to the power of (-5), only x was. Next, we will move y into the numerator and make (-2) into (+2): $$\frac{6}{x^5y^{-2}}\hspace{.25em}» \hspace{.25em}\frac{6y^2}{x^5}$$ $$\frac{6x^{-5}}{y^{-2}}=\frac{6y^2}{x^5}$$

Example 5: Simplify each. $$\require{cancel}\frac{2^5}{2^2}$$ To simplify, keep the base 2 the same and subtract the exponent in the denominator (2) away from the exponent in the numerator (5): $$\frac{2^5}{2^2}=2^{5 - 2}=2^3$$ Let's think about why this works, suppose we just simplified without our rule: $$\frac{2^5}{2^2}=\frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{2 \cdot 2}$$ We can see that we have five factors of 2 in the numerator and two factors of 2 in the denominator. If we cancel common factors, we will be canceling two factors of 2 between the numerator and the denominator. This will leave us with (5 - 2 = 3) three factors of 2 or 8: $$\frac{\cancel{2}\cdot \cancel{2}\cdot 2 \cdot 2 \cdot 2}{\cancel{2}\cdot \cancel{2}}=2 \cdot 2 \cdot 2$$ Example 6: Simplify each. $$\frac{x^{3}}{x^{11}}$$ Keep the base (x) the same and subtract exponents (3 - 11 = -8) $$\frac{x^{3}}{x^{11}}=x^{3 - 11}=x^{-8}$$ We know how to work with negative exponents. The x goes into the denominator and the exponent of (-8) becomes positive: $$x^{-8}=\frac{1}{x^8}$$

### Negative Exponents & the Power of Zero

Up to this point, we have only dealt with whole-number exponents larger than 1. What happens if we see something such as:3

^{-4}What is the value of 3 to the power of (-4)? To understand negative exponents, let's think about a pattern:

3 • 3 • 3 • 3 = 3

^{4}= 813 • 3 • 3 = 3

^{3}= 273 • 3 = 3

^{2}= 93

^{1}= 3Each time we reduce our exponent by 1, we divide by our base of 3. This is because we are removing a factor of 3 when we decrease the exponent by 1. When we go from 3

^{4}(81) to 3^{3}(27), we could just divide 81 by 3 to obtain 27. This pattern continues. If we move to 3^{2}, we can divide 27 by 3 to obtain 9. If we want 3^{1}, we can divide 9 by 3 to obtain 3. So what happens when we get to 3^{0}? We continue the same pattern. We would divide 3 by 3 to obtain 1:3 • 3 • 3 • 3 = 3

^{4}= 813 • 3 • 3 = 3

^{3}= 273 • 3 = 3

^{2}= 93

^{1}= 33

^{0}= 1This works for any non-zero number. We can state that any non-zero number raised to the power of zero is 1. If we try to raise zero to the power of zero, we will have a problem. We can't divide 0 by 0, this is undefined. Therefore, we say zero raised to the power of zero is undefined. What happens if we continue and decrease the exponent by 1 to (-1)? We would continue the pattern. 1 would be divided by 3, and could be written as 1/3:

3 • 3 • 3 • 3 = 3

^{4}= 813 • 3 • 3 = 3

^{3}= 273 • 3 = 3

^{2}= 93

^{1}= 33

^{0}= 13

^{-1}= 1/3As we continue to decrease our exponent by 1, we continue the same process. Divide by the base (3) each time we reduce the exponent by 1:

3 • 3 • 3 • 3 = 3

^{4}= 813 • 3 • 3 = 3

^{3}= 273 • 3 = 3

^{2}= 93

^{1}= 33

^{0}= 13

^{-1}= 1/33

^{-2}= 1/93

^{-3}= 1/273

^{-4}= 1/81Obviously, we will not be going through all this division each time we need to simplify with negative exponents. This was just to give you an understanding of where our simplified result comes from. When we want to simplify with negative exponents, we take the reciprocal of the base and make the exponent positive. Another way to think about this is by stating that we will drag the base and exponent across the fraction bar and make the exponent positive. If we wanted to simplify 3

^{-2}we would take the reciprocal of 3. This would give us 1/3. We would change the exponent from (-2) into (+2): $$3^{-2}=\frac{1}{3^2}=\frac{1}{9}$$ Let's look at a few examples.Example 1: Simplify each. $$5^{-5}$$ Take the reciprocal of the base: $$5^{-5}\hspace{.25em}» \hspace{.25em}\frac{1}{5^{-5}}$$ Make the exponent positive: $$\frac{1}{5^{-5}}\hspace{.25em}» \hspace{.25em}\frac{1}{5^{5}}$$ $$5^{-5}=\frac{1}{5^5}$$ Alternatively, we can start by creating a fraction with 1 as the denominator. When we drag an exponential expression across a fraction bar, the base stays the same and we change the sign of the exponent: $$5^{-5}=\frac{5^{-5}}{1}$$ Now we can drag the 5

^{-5}part into the denominator, we will change our exponent to positive. The numerator will be 1: $$5^{-5}=\frac{5^{-5}}{1}=\frac{1}{5^5}$$ Example 2: Simplify each. $$x^{-11}$$ Take the reciprocal of the base: $$x^{-11}\hspace{.25em}» \hspace{.25em}\frac{1}{x^{-11}}$$ Make the exponent positive: $$\frac{1}{x^{-11}}\hspace{.25em}» \hspace{.25em}\frac{1}{x^{11}}$$ $$x^{-11}=\frac{1}{x^{11}}$$ What happens when we have a negative exponent in the denominator? Again we can use our method where we drag the base across the fraction bar and make the exponent positive. Let's look at an example.Example 3: Simplify each. $$\frac{1}{y^{-4}}$$ Bring the base y into the numerator, the (-4) becomes (+4): $$\frac{1}{y^{-4}}=\frac{y^4}{1}=y^4$$ Example 4: Simplify each. $$\frac{6x^{-5}}{y^{-2}}$$ In this case, we have two exponential expressions with negative exponents. We will move x into the denominator and make (-5) into (+5): $$\frac{6x^{-5}}{y^{-2}}\hspace{.25em}» \hspace{.25em}\frac{6}{x^5y^{-2}}$$ Notice how 6 in the numerator was not moved. The 6 is not raised to the power of (-5), only x was. Next, we will move y into the numerator and make (-2) into (+2): $$\frac{6}{x^5y^{-2}}\hspace{.25em}» \hspace{.25em}\frac{6y^2}{x^5}$$ $$\frac{6x^{-5}}{y^{-2}}=\frac{6y^2}{x^5}$$

### Quotient Rule for Exponents

The quotient rule for exponents allows us to quickly simplify when dividing two exponential expressions that have the same base. The quotient rule for exponents tells us when we divide two numbers or expressions in exponent form and the bases are the same, we can keep the base the same and subtract the exponent in the denominator away from the exponent in the numerator. Let's look at a few examples.Example 5: Simplify each. $$\require{cancel}\frac{2^5}{2^2}$$ To simplify, keep the base 2 the same and subtract the exponent in the denominator (2) away from the exponent in the numerator (5): $$\frac{2^5}{2^2}=2^{5 - 2}=2^3$$ Let's think about why this works, suppose we just simplified without our rule: $$\frac{2^5}{2^2}=\frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{2 \cdot 2}$$ We can see that we have five factors of 2 in the numerator and two factors of 2 in the denominator. If we cancel common factors, we will be canceling two factors of 2 between the numerator and the denominator. This will leave us with (5 - 2 = 3) three factors of 2 or 8: $$\frac{\cancel{2}\cdot \cancel{2}\cdot 2 \cdot 2 \cdot 2}{\cancel{2}\cdot \cancel{2}}=2 \cdot 2 \cdot 2$$ Example 6: Simplify each. $$\frac{x^{3}}{x^{11}}$$ Keep the base (x) the same and subtract exponents (3 - 11 = -8) $$\frac{x^{3}}{x^{11}}=x^{3 - 11}=x^{-8}$$ We know how to work with negative exponents. The x goes into the denominator and the exponent of (-8) becomes positive: $$x^{-8}=\frac{1}{x^8}$$

#### Skills Check:

Example #1

Simplify each. $$\frac{\left(x^{2}\right)^{3}}{(2x^{4}y)^{-4}\cdot 2xy^{-3}}$$

Please choose the best answer.

A

$$\frac{y^{12}}{2x^{10}}$$

B

$$\frac{xy^{10}}{32}$$

C

$$8x^{21}y^{7}$$

D

$$\frac{y^{6}}{2x^{2}}$$

E

$$\frac{8x^{2}}{y^{4}}$$

Example #2

Simplify each. $$\frac{2u^{3}v^{0}}{vu^{3}\cdot 2u^{-4}v^{2}\cdot (v^{3})^{-1}}$$

Please choose the best answer.

A

$$u^{4}$$

B

$$\frac{1}{u^{9}v^{6}}$$

C

$$\frac{8u^{3}}{v^{6}}$$

D

$$8v^{2}u^{6}$$

E

$$1$$

Example #3

Simplify each. $$\frac{xy^{-3}\cdot (x^{-2}y^{4})^{4}}{x^{4}}$$

Please choose the best answer.

A

$$y^{4}$$

B

$$\frac{16y^{7}}{x^{2}}$$

C

$$\frac{y^{13}}{x^{11}}$$

D

$$4x^{15}y^{4}$$

E

$$16x^{5}y^{3}$$

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