# 8.1: Exponential Properties Involving Products

Difficulty Level: Basic Created by: CK-12
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What if you rolled a die and got an integer between 1 to 6? Try multiplying the square of the integer by the cube of the integer. Now take the square of the result. If the original integer were represented by the variable \begin{align*}x\end{align*}, what would you have after you performed these operations? How would you find the exponent in your answer? In this Concept, you'll learn how to simplify exponential expressions by using the exponential properties involving products so that you can correctly solve this type of problem.

### Exponential Properties Involving Products

In this Concept, you will be learning what an exponent is and about the properties and rules of exponents. You will also learn how to use exponents in problem solving.

Definition: An exponent is a power of a number that shows how many times that number is multiplied by itself.

An example would be \begin{align*}2^3\end{align*}. You would multiply 2 by itself 3 times: \begin{align*}2 \times 2 \times 2\end{align*}. The number 2 is the base and the number 3 is the exponent. The value \begin{align*}2^3\end{align*} is called the power.

#### Example A

Write in exponential form: \begin{align*}\alpha \times \alpha \times \alpha \times \alpha\end{align*}.

Solution: You must count the number of times the base, \begin{align*}\alpha\end{align*}, is being multiplied by itself. It’s being multiplied four times so the solution is \begin{align*}\alpha^4\end{align*}.

Note: There are specific rules you must remember when taking powers of negative numbers.

\begin{align*}(negative \ number) \times (positive \ number) &= negative \ number\\ (negative \ number) \times (negative \ number) &= positive \ number\end{align*}

For even powers of negative numbers, the answer will always be positive. Pairs can be made with each number and the negatives will be cancelled out.

\begin{align*}(-2)^4 = (-2)(-2)(-2)(-2) = (-2)(-2) \cdot (-2)(-2) = +16\end{align*}

For odd powers of negative numbers, the answer is always negative. Pairs can be made but there will still be one negative number unpaired, making the answer negative.

\begin{align*}(-2)^5 = (-2)(-2)(-2)(-2)(-2) = (-2)(-2) \cdot (-2)(-2) \cdot (-2) = -32\end{align*}

When we multiply the same numbers, each with different powers, it is easier to combine them before solving. This is why we use the Product of Powers Property.

Product of Powers Property: For all real numbers \begin{align*}\chi, \chi^n \cdot \chi^m = \chi^{n+m}\end{align*}.

#### Example B

Multiply \begin{align*}\chi^4 \cdot \chi^5\end{align*}.

Solution: \begin{align*}\chi^4 \cdot \chi^5 = \chi^{4+5} = \chi^9\end{align*}

Note that when you use the product rule you DO NOT MULTIPLY BASES.

#### Example C

\begin{align*}2^2 \cdot 2^3 \neq 4^5\end{align*}

Another note is that this rule APPLIES ONLY TO TERMS THAT HAVE THE SAME BASE.

The Power of a Product

\begin{align*}& (x^4)^3 = x^4 \cdot x^4 \cdot x^4 \qquad \qquad 3 \ \text{factors of} \ x \ \text{to the power} \ 4.\\ & \underbrace{(x \cdot x \cdot x \cdot x}_{x^4}) \cdot \underbrace{(x \cdot x \cdot x \cdot x}_{x^4}) \cdot \underbrace{(x \cdot x \cdot x \cdot x}_{x^4})=\underbrace{(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x)}_{x^{12}}\end{align*}

This situation is summarized below.

Power of a Product Property: For all real numbers \begin{align*}\chi\end{align*}:

\begin{align*}(\chi^n)^m = \chi^{n \cdot m}\end{align*}

The Power of a Product Property is similar to the Distributive Property. Everything inside the parentheses must be taken to the power outside. For example, \begin{align*}(x^2y)^4=(x^2)^4 \cdot (y)^4=x^8y^4\end{align*}. Watch how it works the long way.

\begin{align*}\underbrace{(x \cdot x \cdot y)}_{x^2y} \cdot \underbrace{(x \cdot x \cdot y)}_{x^2y} \cdot \underbrace{(x \cdot x \cdot y)}_{x^2y} \cdot \underbrace{(x \cdot x \cdot y)}_{x^2y}=\underbrace{(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y)}_{x^8y4}\end{align*}

The Power of a Product Property does not work if you have a sum or difference inside the parenthesis. For example, \begin{align*}(\chi+\gamma)^2 \neq \chi^2 + \gamma^2\end{align*}. Because it is an addition equation, it should look like \begin{align*}(\chi+\gamma)(\chi+\gamma)\end{align*}.

#### Example D

Simplify \begin{align*}(\chi^2)^3\end{align*}.

Solution: \begin{align*}(\chi^2)^3= \chi^{2\cdot 3} = \chi^6\end{align*}

### Guided Practice

1. Show that \begin{align*}2^2 \cdot 3^3 \neq 6^5\end{align*}.

2. Simplify \begin{align*}(\chi^3\cdot \chi^4)^2\end{align*}.

Solutions:

1. Evaluate each side separately, to show that they are not equal:

\begin{align*}2^2 \cdot 3^3 &= (2\cdot 2)\cdot (3\cdot 3 \cdot 3)=4\cdot 27=108 \\ 6^5 &= 6\cdot 6\cdot 6\cdot 6\cdot 6=7776 \end{align*}

Since \begin{align*}108 \neq 7776\end{align*}, this means that \begin{align*}2^2 \cdot 3^3 \neq 6^5\end{align*}.

2. \begin{align*}\left(\chi^3\cdot \chi^4\right)^2=\left(\chi^{3+4}\right)^2=\left(\chi^7\right)^2=\chi^{7\cdot 2}=\chi^{14} \end{align*}

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Exponent Properties Involving Products (14:00)

1. Consider \begin{align*}a^5\end{align*}. a. What is the base? b. What is the exponent? c. What is the power? d. How can this power be written using repeated multiplication?

Determine whether the answer will be positive or negative. You do not have to provide the answer.

1. \begin{align*}-(3^4)\end{align*}
2. \begin{align*}-8^2\end{align*}
3. \begin{align*}10 \times (-4)^3\end{align*}
4. What is the difference between \begin{align*}-5^2\end{align*} and \begin{align*}(-5)^2\end{align*}?

Write in exponential notation.

1. \begin{align*}2 \cdot 2\end{align*}
2. \begin{align*}(-3)(-3)(-3)\end{align*}
3. \begin{align*}y \cdot y \cdot y \cdot y \cdot y\end{align*}
4. \begin{align*}(3a)(3a)(3a)(3a)\end{align*}
5. \begin{align*}4 \cdot 4 \cdot 4 \cdot 4 \cdot 4\end{align*}
6. \begin{align*}3x \cdot 3x \cdot 3x\end{align*}
7. \begin{align*}(-2a)(-2a)(-2a)(-2a)\end{align*}
8. \begin{align*}6 \cdot 6 \cdot 6 \cdot x \cdot x \cdot y \cdot y \cdot y \cdot y\end{align*}

Find each number.

1. \begin{align*}1^{10}\end{align*}
2. \begin{align*}0^3\end{align*}
3. \begin{align*}7^3\end{align*}
4. \begin{align*}-6^2\end{align*}
5. \begin{align*}5^4\end{align*}
6. \begin{align*}3^4 \cdot 3^7\end{align*}
7. \begin{align*}2^6 \cdot 2\end{align*}
8. \begin{align*}(4^2)^3\end{align*}
9. \begin{align*}(-2)^6\end{align*}
10. \begin{align*}(0.1)^5\end{align*}
11. \begin{align*}(-0.6)^3\end{align*}

Multiply and simplify.

1. \begin{align*}6^3 \cdot 6^6\end{align*}
2. \begin{align*}2^2 \cdot 2^4 \cdot 2^6\end{align*}
3. \begin{align*}3^2 \cdot 4^3\end{align*}
4. \begin{align*}x^2 \cdot x^4\end{align*}
5. \begin{align*}x^2 \cdot x^7\end{align*}
6. \begin{align*}(y^3)^5\end{align*}
7. \begin{align*}(-2 y^4)(-3y)\end{align*}
8. \begin{align*}(4a^2)(-3a)(-5a^4)\end{align*}

Simplify.

1. \begin{align*}(a^3)^4\end{align*}
2. \begin{align*}(xy)^2\end{align*}
3. \begin{align*}(3a^2b^3)^4\end{align*}
4. \begin{align*}(-2xy^4z^2)^5\end{align*}
5. \begin{align*}(3x^2 y^3) \cdot (4xy^2)\end{align*}
6. \begin{align*}(4xyz) \cdot (x^2y^3) \cdot (2yz^4)\end{align*}
7. \begin{align*}(2a^3b^3)^2\end{align*}
8. \begin{align*}(-8x)^3(5x)^2\end{align*}
9. \begin{align*}(4a^2)(-2a^3)^4\end{align*}
10. \begin{align*}(12xy)(12xy)^2\end{align*}
11. \begin{align*}(2xy^2)(-x^2y)^2(3x^2y^2)\end{align*}

Mixed Review

1. How many ways can you choose a 4-person committee from seven people?
2. Three canoes cross a finish line to earn medals. Is this an example of a permutation or a combination? How many ways are possible?
3. Find the slope between (–9, 11) and (18, 6).
4. Name the number set(s) to which \begin{align*}\sqrt{36}\end{align*} belongs.
5. Simplify \begin{align*}\sqrt{74x^2}\end{align*}.
6. 78 is 10% of what number?
7. Write the equation for the line containing (5, 3) and (3, 1).

### Notes/Highlights Having trouble? Report an issue.

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### Vocabulary Language: English Spanish

TermDefinition
Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
Power of a Product Property For all real numbers $\chi$, $(\chi^n)^m = \chi^{n \cdot m}$.
Product of Powers Property For all real numbers $\chi$, $\chi^n \cdot \chi^m = \chi^{n+m}$.
Base When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression $32^4$, 32 is the base, and 4 is the exponent.
Power The "power" refers to the value of the exponent. For example, $3^4$ is "three to the fourth power".

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