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Exponent Properties with Variable Expressions

Operations with exponents, including negative and fractional exponents

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Product and Quotient Properties of Exponents

Miguel says that the expression \begin{align*}\frac{2^5 \cdot 2^4}{2^2}\end{align*} equals \begin{align*}2^{10}\end{align*}.

Alise says that it is equal to \begin{align*}2^7\end{align*}.

Carlos says that because the exponents of the terms are different, the expression can't be simplified.

Which one of them is correct?

Product and Quotient Properties of Exponents

To review, the power (or exponent) of a number is the little number in the superscript. The number that is being raised to the power is called the base. The exponent indicates how many times the base is multiplied by itself.

There are several properties of exponents. We will investigate two in this concept.

Let's expand and solve \begin{align*}5^6\end{align*}.

 \begin{align*}5^6\end{align*} means 5 times itself six times.

\begin{align*}5^6 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5 = 15,625\end{align*}

Product Property

Step 1: Expand \begin{align*}3^4 \cdot 3^5\end{align*}.

\begin{align*}\underbrace{3 \cdot 3 \cdot 3 \cdot 3}_{3^4} \cdot \underbrace{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}_{3^5}\end{align*}

Step 2: Rewrite this expansion as one power of three.


Step 3: What is the sum of the exponents?

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

Step 4: Fill in the blank: \begin{align*}a^m \cdot a^n = a^{-^+-}\end{align*}

\begin{align*}a^m \cdot a^n = a^{m+n}\end{align*}

Rather than expand the exponents every time or find the powers separately, we can use this property to simplify the product of two exponents with the same base.

Let's simplify the following expressions using the Product Property above.

  1. \begin{align*}x^3 \cdot x^8\end{align*}

\begin{align*}x^3 \cdot x^8 = x^{3+8} = x^{11}\end{align*}

  1. \begin{align*}xy^2 x^2 y^9\end{align*}

If a number does not have an exponent, you may assume the exponent is 1. Reorganize this expression so the \begin{align*}x\end{align*}’s are together and \begin{align*}y\end{align*}’s are together.

\begin{align*}xy^2 x^2 y^9 = x^1 \cdot x^2 \cdot y^2 \cdot y^9 = x^{1+2} \cdot y^{2+9} = x^3 y^{11}\end{align*}

Quotient Property

Step 1: Expand \begin{align*}2^8 \div 2^3\end{align*}. Also, rewrite this as a fraction.

\begin{align*}\frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{2 \cdot 2 \cdot 2}\end{align*}

Step 2: Cancel out the common factors and write the answer one power of 2.

\begin{align*}\frac{\cancel{2} \cdot \cancel{2} \cdot \cancel{2} \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{\cancel{2} \cdot \cancel{2} \cdot \cancel{2}} = 2^5\end{align*}

Step 3: What is the difference of the exponents?

\begin{align*}8 - 3 = 5\end{align*}

Step 4: Fill in the blank: \begin{align*}\frac{a^m}{a^n} = a^{-^--}\end{align*}

\begin{align*}\frac{a^m}{a^n} = a^{m-n}\end{align*}

Let's simplify the following expressions using the Quotient Property above.

  1. \begin{align*}\frac{5^9}{5^7}\end{align*}

\begin{align*}\frac{5^9}{5^7} = 5^{9-7} = 5^2 = 25\end{align*}

  1. \begin{align*}\frac{x^4}{x^2}\end{align*}

\begin{align*}\frac{x^4}{x^2} = x^{4-2} = x^2\end{align*}


Example 1

Earlier, you were asked to find which student is correct.

Using the Product Property and then the Quotient Property, the expression can be simplified.

\begin{align*}\frac{2^5 \cdot 2^4}{2^2}\\ = \frac{2^9}{2^2}\\ = 2^7\end{align*}

Therefore, Alise is correct.

Simplify the following expressions. Evaluate any numerical answers.

Example 2

\begin{align*}7 \cdot 7^2\end{align*}

\begin{align*}7 \cdot 7^2 = 7^{1+2} = 7^3 = 343\end{align*}

Example 3


\begin{align*}\frac{3^7}{3^3} = 3^{7-3} = 3^4 = 81\end{align*}

Example 4

\begin{align*}\frac{16x^4 y^5}{4x^2 y^2}\end{align*}

\begin{align*}\frac{16x^4 y^5}{4x^2 y^2} = 4x^{4-2} y^{5-3} = 4x^2 y^2\end{align*}


Expand the following numbers and evaluate.

  1. \begin{align*}2^6\end{align*}
  2. \begin{align*}10^3\end{align*}
  3. \begin{align*}(-3)^5\end{align*}
  4. \begin{align*}(0.25)^4\end{align*}

Simplify the following expressions. Evaluate any numerical answers.

  1. \begin{align*}4^2 \cdot 4^7\end{align*}
  2. \begin{align*}6 \cdot 6^3 \cdot 6^2\end{align*}
  3. \begin{align*}\frac{8^3}{8}\end{align*}
  4. \begin{align*}\frac{2^4 \cdot 3^5}{2 \cdot 3^2}\end{align*}
  5. \begin{align*}b^6 \cdot b^3\end{align*}
  6. \begin{align*}5^2 x^4 \cdot x^9\end{align*}
  7. \begin{align*}\frac{y^{12}}{y^5}\end{align*}
  8. \begin{align*}\frac{a^8 \cdot b^6}{b \cdot a^4}\end{align*}
  9. \begin{align*}\frac{3^7 x^6}{3^3 x^3}\end{align*}
  10. \begin{align*}d^5 f^3 d^9 f^7\end{align*}
  11. \begin{align*}\frac{2^8 m^{18} n^{14}}{2^5 m^{11} n^4}\end{align*}
  12. \begin{align*}\frac{9^4 p^5 q^8}{9^2 pq^2}\end{align*}

Investigation Evaluate the powers of negative numbers.

  1. Find:
    1. \begin{align*}(-2)^1\end{align*}
    2. \begin{align*}(-2)^2\end{align*}
    3. \begin{align*}(-2)^3\end{align*}
    4. \begin{align*}(-2)^4\end{align*}
    5. \begin{align*}(-2)^5\end{align*}
    6. \begin{align*}(-2)^6\end{align*}
  2. Make a conjecture about even vs. odd powers with negative numbers.
  3. Is \begin{align*}(-2)^4\end{align*} different from \begin{align*}-2^4\end{align*}? Why or why not?

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.1. 

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Exponent Exponents are used to describe the number of times that a term is multiplied by itself.
Power The "power" refers to the value of the exponent. For example, 3^4 is "three to the fourth power".
power to a power Power to a power is a number raised to an exponent which in turn is raised to another exponent.
Product of Powers Property The product of powers property states that a^m \cdot a^n = a^{m+n}.
Quotients of Powers Property The quotient of powers property states that \frac{a^m}{a^n} = a^{m-n} for a \ne 0.

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