Lesson Objectives

- Learn how to simplify expressions using the product rule for exponents
- Learn how to simplify expressions using the power rules for exponents
- Learn how to work with an exponent of zero

## Product & Power Rules for Exponents

In our pre-algebra course, we learned how to use whole-number exponents larger than 1 to show the repeated multiplication of the same number. When we work with exponents, we have a large number known as a base and a small number known as the exponent. The base represents the number being multiplied by itself in the repeated multiplication. A whole-number exponent larger than 1 represents the number of factors of the base. Let's look at an example.

Example 1: Write each repeated multiplication using exponential form.

9 • 9 • 9 • 9

9 • 9 • 9 • 9 = 9

We have 4 factors of 9. The 9 is the number being multiplied by itself in the repeated multiplication. This will be used as our larger number or our base. Since we have 4 factors of 9, our exponent or smaller number will be 4.

We should also know how to reverse the process and write a number in exponent form as repeated multiplication. Let's look at an example.

Example 2: Write each as repeated multiplication.

18

18

We have 18 raised to the 3rd power. Our base, 18, is the number being multiplied by itself. Our exponent, 3, is the number of factors of 18.

Example 3: Simplify each.

4

Notice how the base (4) is the same in each case. We must have the same base to use this rule. Our product rule for exponents tells us to keep the base the same and add exponents:

4

How can we prove this is true? It's actually pretty simple when we show the multiplication process:

4

Since 4

Example 4: Simplify each.

x

Keep the base (x) the same and add exponents (9 + 12 = 21):

x

Example 5: Simplify each.

9

Keep the base (9) the same and add exponents (4 + 11 = 15):

9

(3

This tells us that 3

(3

We know from our product rule for exponents that we can keep the base (3) the same and add the exponents (2 + 2 + 2 = 6):

3

How could we get this result more quickly? We could simply have kept the base the same and multiplied our exponents:

(3

This is exactly what the power to power rule tells us to do. When we have a power raised to another power, the power to power rule states that we keep our base the same and we multiply the exponents. Let's look at some examples.

Example 6: Simplify each.

(14

Keep the base (14) the same and multiply the exponents (4 • 11 = 44):

(14

Example 7: Simplify each.

(x

Keep the base (x) the same and multiply exponents (23 • 5 = 115):

(x

Example 8: Simplify each.

(3 • 2)

According to our rule, we can raise each factor (3 and 2) to the power of 4:

(3 • 2)

We can check to see if this is true, let's start by working inside of the parentheses in the original problem:

(3 • 2)

Now let's look at the alternative:

3

It works the same either way. You might think this property is useless based on this example, simplifying in the parentheses would have been faster here. In many cases, we will use this property with variables when it is not possible to simplify inside of the parentheses. Let's look at another example.

Example 9: Simplify each.

(5xy)

We will raise each factor to the power of 9:

(5xy)

Example 10: Simplify each.

$$\left(\frac{x}{3}\right)^7$$ We will raise the numerator (x) and the denominator (3) to the 7th power: $$\left(\frac{x}{3}\right)^7=\frac{x^7}{3^7}$$

3

What is the value of 3 to the power of (0)? To deal with an exponent of zero, 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 nonzero number. We can state that any nonzero 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.

Example 11: Simplify each.

$$(2x^4)^0, x≠0$$ Raising any nonzero number to the power of zero gives us 1 as a result. The same is true for a nonzero expression. In this case, we have stated that x is not zero, therefore, the expression 2x

Example 1: Write each repeated multiplication using exponential form.

9 • 9 • 9 • 9

9 • 9 • 9 • 9 = 9

^{4}We have 4 factors of 9. The 9 is the number being multiplied by itself in the repeated multiplication. This will be used as our larger number or our base. Since we have 4 factors of 9, our exponent or smaller number will be 4.

We should also know how to reverse the process and write a number in exponent form as repeated multiplication. Let's look at an example.

Example 2: Write each as repeated multiplication.

18

^{3}18

^{3}= 18 • 18 • 18We have 18 raised to the 3rd power. Our base, 18, is the number being multiplied by itself. Our exponent, 3, is the number of factors of 18.

### Product Rule for Exponents

The product rule for exponents allows us to quickly simplify products of numbers or expressions in exponent form. The product rule for exponents states when we multiply two numbers or expressions in exponent form with the same base, we keep the base the same and add exponents. Let's look at a few examples.Example 3: Simplify each.

4

^{2}• 4^{3}Notice how the base (4) is the same in each case. We must have the same base to use this rule. Our product rule for exponents tells us to keep the base the same and add exponents:

4

^{2}• 4^{3}= 4^{2 + 3}= 4^{5}How can we prove this is true? It's actually pretty simple when we show the multiplication process:

4

^{2}• 4^{3}= 4 • 4 • 4 • 4 • 4 = 4^{5}Since 4

^{2}has two factors of 4 and 4^{3}has three factors of 4, when they are multiplied together we end up with (2 + 3 = 5) five factors of 4. Therefore, we can keep our base the same and just add the exponents.Example 4: Simplify each.

x

^{9}• x^{12}Keep the base (x) the same and add exponents (9 + 12 = 21):

x

^{9}• x^{12}= x^{9 + 12}= x^{21}Example 5: Simplify each.

9

^{4}• 9^{11}Keep the base (9) the same and add exponents (4 + 11 = 15):

9

^{4}• 9^{11}= 9^{4 + 11}= 9^{15}### Power to Power Rule for Exponents

We will also come across power rules for exponents. The power to power rule is used when we have a power raised to another power. As an example, suppose we saw the following scenario:(3

^{2})^{3}This tells us that 3

^{2}(three squared) is raised to the 3rd power. In other words, we have:(3

^{2})^{3}= 3^{2}• 3^{2}• 3^{2}We know from our product rule for exponents that we can keep the base (3) the same and add the exponents (2 + 2 + 2 = 6):

3

^{2}• 3^{2}• 3^{2}= 3^{6}How could we get this result more quickly? We could simply have kept the base the same and multiplied our exponents:

(3

^{2})^{3}= 3^{2 • 3}= 3^{6}This is exactly what the power to power rule tells us to do. When we have a power raised to another power, the power to power rule states that we keep our base the same and we multiply the exponents. Let's look at some examples.

Example 6: Simplify each.

(14

^{4})^{11}Keep the base (14) the same and multiply the exponents (4 • 11 = 44):

(14

^{4})^{11}= 14^{44}Example 7: Simplify each.

(x

^{23})^{5}Keep the base (x) the same and multiply exponents (23 • 5 = 115):

(x

^{23})^{5}= x^{115}### Power of a Product Rule for Exponents

Next on our list of power rules comes the power of a product rule. This rule states that we can raise a product to a power by raising each factor to the power. Let's look at an example.Example 8: Simplify each.

(3 • 2)

^{4}According to our rule, we can raise each factor (3 and 2) to the power of 4:

(3 • 2)

^{4}= 3^{4}• 2^{4}We can check to see if this is true, let's start by working inside of the parentheses in the original problem:

(3 • 2)

^{4}= 6^{4}= 1296Now let's look at the alternative:

3

^{4}• 2^{4}= 81 • 16 = 1296It works the same either way. You might think this property is useless based on this example, simplifying in the parentheses would have been faster here. In many cases, we will use this property with variables when it is not possible to simplify inside of the parentheses. Let's look at another example.

Example 9: Simplify each.

(5xy)

^{9}We will raise each factor to the power of 9:

(5xy)

^{9}= 5^{9}x^{9}y^{9}### Power of a Quotient Rule for Exponents

Similar to our last rule, we can raise a quotient to a power by raising the numerator (dividend) and denominator (divisor) to the power. Let's take a look at an example.Example 10: Simplify each.

$$\left(\frac{x}{3}\right)^7$$ We will raise the numerator (x) and the denominator (3) to the 7th power: $$\left(\frac{x}{3}\right)^7=\frac{x^7}{3^7}$$

### An Exponent of Zero

What happens if we see something such as:3

^{0}What is the value of 3 to the power of (0)? To deal with an exponent of zero, 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 nonzero number. We can state that any nonzero 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.

Example 11: Simplify each.

$$(2x^4)^0, x≠0$$ Raising any nonzero number to the power of zero gives us 1 as a result. The same is true for a nonzero expression. In this case, we have stated that x is not zero, therefore, the expression 2x

^{4}will never be zero. When we raise this expression to the power of zero, the result will be 1. $$(2x^4)^0=1, x≠0$$#### Skills Check:

Example #1

Simplify each: $$x^4 \cdot x^7$$

Please choose the best answer.

A

$$x^{28}$$

B

$$x^{11}$$

C

$$2x^{11}$$

D

$$2x^{28}$$

E

$$x^3$$

Example #2

Simplify each: $$(5x^2)^3$$

Please choose the best answer.

A

$$5x^5$$

B

$$125x^5$$

C

$$125x^6$$

D

$$5x^6$$

E

$$5x^{30}$$

Example #3

Simplify each: $$(x^2y^2)^0 \cdot (x^5y^3)^2$$

Please choose the best answer.

A

$$x^{14}y^{12}$$

B

$$x^{20}y^{12}$$

C

$$2x^{10}y^{6}$$

D

$$x^{10}y^6$$

E

$$1$$

Example #4

Simplify each: $$(x^3)^4 \cdot (x^7 y^4)^5$$

Please choose the best answer.

A

$$x^{19}y^9$$

B

$$x^{47}y^{20}$$

C

$$12x^{13}y^{19}$$

D

$$2x^{47}y^{20}$$

E

$$2x^{42}y^{20}$$

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