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Exponential Properties Involving Products

Add exponents to multiply exponents by other exponents

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Exponential Properties Involving Products

Suppose you rolled a die and got an integer between 1 and 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?

Exponential Properties Involving Products

Exponential Form

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.

Let's write the following expression in exponential form:

\begin{align*}\alpha \times \alpha \times \alpha \times \alpha\end{align*}.

Exponential form is this expression written with an exponent. 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 that 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*}

Product of Powers

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.

The Product of Powers Property states that for all real numbers \begin{align*}\chi, \chi^n \cdot \chi^m = \chi^{n+m}\end{align*}.

Let's multiply the following expressions:

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

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

Note that the Product of Powers Property applies only to terms that have the same base.

  1.  \begin{align*}2^2\cdot 2^3\end{align*} 

Note that \begin{align*}2^2 \cdot 2^3 \neq 4^5\end{align*}. You cannot multiply the bases.

\begin{align*}2^2 \cdot 2^3 = 2 ^{2+3} = 2^5=32\end{align*} 

Power of a Power

The Product of Powers Property extends to the Power of a Power Property. 

The Power of a Power Property states that for all real numbers \begin{align*}\chi, \space{(\chi^n)}^m = \chi^{nm}\end{align*}.

The Power of a Power 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 Power 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*}.

Let's simplify the following expressions: 

  1.  \begin{align*}{(x^4)}^3\end{align*}

\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.\\ & = x^{4+4+4}\\ &=x^{12}\end{align*}

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

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


Example 1

Earlier, you were told to roll a die to get an integer between 1 and 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?

Let's call the integer that we roll \begin{align*}x\end{align*}. We want to multiply the square of the integer by the cube of the integer. This gives us the expression: 

\begin{align*}x^2 \cdot x^3 = x^5\end{align*} 

Then, we need to square the result:

\begin{align*}(x^5)^2= x^{10}\end{align*}

In the first step, the Product of Powers Property was used and in the second step, the Power of a Power Property was used.

The final result is \begin{align*}x^{10}\end{align*}

Example 2

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

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*}.

Example 3

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

\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*}


  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*}


  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).

Review (Answer)

To see the Review answers, open this PDF file and look for section 8.1. 

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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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