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8.1: Exponential Properties Involving Products

Difficulty Level: At Grade Created by: CK-12

In this lesson, 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*}23. You would multiply 2 by itself 3 times, \begin{align*}2 \times 2 \times 2\end{align*}2×2×2. The number 2 is the base and the number 3 is the exponent. The value \begin{align*}2^3\end{align*}23 is called the power.

Example 1: Write in exponential form: \begin{align*}\alpha \times \alpha \times \alpha \times \alpha\end{align*}α×α×α×α.

Solution: You must count the number of times the base, \begin{align*}\alpha\end{align*}α, is being multiplied by itself. It’s being multiplied four times so the solution is \begin{align*}\alpha^4\end{align*}α4.

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*}

(negative number)×(positive number)(negative number)×(negative number)=negative number=positive number

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*}χ,χnχm=χn+m.

Example 2: Multiply \begin{align*}\chi^4 \cdot \chi^5\end{align*}χ4χ5.

Solution: \begin{align*}\chi^4 \cdot \chi^5 = \chi^{4+5} = \chi^9\end{align*}χ4χ5=χ4+5=χ9

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

Example: \begin{align*}2^2 \cdot 2^3 \neq 4^5\end{align*}222345

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

Example: \begin{align*}2^2 \cdot 3^3 \neq 6^5\end{align*}223365

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

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

Practice Set

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*}. 1. What is the base? 2. What is the exponent? 3. What is the power? 4. 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*}


  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).

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