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Zero, Negative, and Fractional Exponents

Evaluate constant and variable terms with negative exponents

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Negative and Zero Exponents

The magnitude of an earthquake represents the exponent m in the expression \begin{align*}10^m.\end{align*}10m.

Valdivia, Chile has suffered two major earthquakes. The 1575 Valdivia earthquake had a magnitude of 8.5. The world's largest earthquake was the 1960 Valdivia earthquake at a magnitude of 9.5.

What was the size of the 1575 earthquake compared to the 1960 one?

Source: Wikipedia

Negative and Zero Exponents

In this concept, we will introduce negative and zero exponents.

Zero Exponents

Step 1: Evaluate \begin{align*}\frac{5^6}{5^6}\end{align*}5656 by using the Quotient of Powers property.

\begin{align*}\frac{5^6}{5^6} = 5^{6-6} = 5^0\end{align*}5656=566=50

Step 2: What is a number divided by itself? Apply this to Step 1.

\begin{align*}\frac{5^6}{5^6} = 1\end{align*}5656=1

Step 3: Fill in the blanks. \begin{align*}\frac{a^m}{a^m} = a^{m-m} = a^- = _-\end{align*}amam=amm=a=

\begin{align*}a^0 = 1\end{align*}a0=1

Negative Exponents

Step 1: Expand \begin{align*}\frac{3^2}{3^7}\end{align*}3237 and cancel out the common 3’s and write your answer with positive exponents.

\begin{align*}\frac{3^2}{3^7} = \frac{\cancel{3} \cdot \cancel{3}}{\cancel{3} \cdot \cancel{3} \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3} = \frac{1}{3^5}\end{align*}

Step 2: Evaluate \begin{align*}\frac{3^2}{3^7}\end{align*} by using the Quotient of Powers property.

\begin{align*}\frac{3^2}{3^7} = 3^{2-7} = 3^{-5}\end{align*}

Step 3: Are the answers from Step 1 and Step 2 equal? Write them as a single statement.

\begin{align*}\frac{1}{3^5} = 3^{-5}\end{align*}

Step 4: Fill in the blanks. \begin{align*}\frac{1}{a^m} = a-\end{align*} and \begin{align*}\frac{1}{a^{-m}} = a-\end{align*}

\begin{align*}\frac{1}{a^m} = a^{-m}\end{align*} and \begin{align*}\frac{1}{a^{-m}} = a^m\end{align*}

From the the discussions above, we have learned two very important properties of exponents. First, anything to the zero power is one. Second, negative exponents indicate placement. If an exponent is negative, it needs to be moved from where it is to the numerator or denominator. 

Let's simplify the following expressions. Your answer should only have positive exponents.

  1. \begin{align*}\frac{5^2}{5^5}\end{align*}

\begin{align*}\frac{5^2}{5^5} = 5^{-3} = \frac{1}{5^3} = \frac{1}{125}\end{align*}

  1. \begin{align*}\frac{x^7 yz^{12}}{x^{12} yz^7}\end{align*}

\begin{align*}\frac{x^7 yz^{12}}{x^{12} yz^7} = \frac{y^{1-1} z^{12-7}}{x^{12-7}} = \frac{y^0 z^5}{x^5} = \frac{z^5}{x^5}\end{align*}

  1. \begin{align*}\frac{a^4 b^0}{a^8 b}\end{align*}

\begin{align*}\frac{a^4 b^0}{a^8 b} = a^{4-8} b^{0-1} = a^{-4} b^{-1} = \frac{1}{a^4 b}\end{align*}


\begin{align*}\frac{a^4 b^0}{a^8 b} = \frac{1}{a^{8-4} b} = \frac{1}{a^4 b}\end{align*}

  1. \begin{align*}\frac{xy^5}{8y^{-3}}\end{align*}

\begin{align*}\frac{xy^5}{8y^{-3}} = \frac{xy^5 y^3}{8} = \frac{xy^{5+3}}{8} = \frac{xy^8}{8}\end{align*}

  1. \begin{align*}\frac{27 g^{-7} h^0}{18 g}\end{align*}

\begin{align*}\frac{27 g^{-7} h^0}{18 g} = \frac{3}{2g^1 g^7} = \frac{3}{2g^{1+7}} = \frac{3}{2g^8}\end{align*}

Now, let's multiply the following two fractions together and simplify. The answer should only have positive exponents.

\begin{align*}\frac{4x^{-2} y^5}{20x^8} \cdot \frac{-5x^6 y}{15y^{-9}}\end{align*}

The easiest way to approach this problem is to multiply the two fractions together first and then simplify.

\begin{align*}\frac{4x^{-2} y^5}{20x^8} \cdot \frac{-5x^6 y}{15y^{-9}} = -\frac{20x^{-2+6} y^{5+1}}{300x^8 y^{-9}} = -\frac{x^{-2+6-8}y^{5+1+9}}{15} = -\frac{x^{-4} y^{15}}{15} = -\frac{y^{15}}{15x^4}\end{align*}


Example 1

Earlier, you were asked to find the size of the 1575 earthquake compared to the 1960 earthquake. 

Set each earthquake's magnitude up as an exponential expression and divide.

\begin{align*}\frac{10^{8.5}}{10^{9.5}}\\ = 10^{-1}\\ = \frac{1}{10^1}\\ = \frac{1}{10}\end{align*}

Therefore, the size of the 1575 earthquake was \begin{align*}\frac{1}{10}\end{align*} the 1960 one.

Simplify the following expressions.

Example 2


\begin{align*}\frac{8^6}{8^9} = 8^{6-9} = \frac{1}{8^3} = \frac{1}{512}\end{align*}

Example 3

\begin{align*}\frac{3x^{10} y^2}{21x^7 y^{-4}}\end{align*}

\begin{align*}\frac{3x^{10} y^2}{21x^7 y^{-4}} = \frac{x^{10-7} y^{2-(-4)}}{7} = \frac{x^3 y^6}{7}\end{align*}

Example 4

\begin{align*}\frac{2a^8 b^{-4}}{16a^{-5}} \cdot \frac{4^3 a^{-3} b^0}{a^4 b^7}\end{align*}

\begin{align*}\frac{2a^8 b^{-4}}{16a^{-5}} \cdot \frac{4^3 a^{-3} b^0}{a^4 b^7} = \frac{128a^{8-3} b^{-4}}{16a^{-5+4} b^7} = \frac{8a^{5+1}}{b^{7+4}} = \frac{8a^6}{b^{11}}\end{align*}


Simplify the following expressions. Answers cannot have negative exponents.

  1. \begin{align*}\frac{8^2}{8^4}\end{align*}
  2. \begin{align*}\frac{x^6}{x^{15}}\end{align*}
  3. \begin{align*}\frac{7^{-3}}{7^{-2}}\end{align*}
  4. \begin{align*}\frac{y^{-9}}{y^{10}}\end{align*}
  5. \begin{align*}\frac{x^0 y^5}{xy^7}\end{align*}
  6. \begin{align*}\frac{a^{-1} b^8}{a^5 b^7}\end{align*}
  7. \begin{align*}\frac{14c^{10} d^{-4}}{21c^6 d^{-3}}\end{align*}
  8. \begin{align*}\frac{8g^0 h}{30g^{-9} h^2}\end{align*}
  9. \begin{align*}\frac{5x^4}{10y^{-2}} \cdot \frac{y^7 x}{x^{-1} y}\end{align*}
  10. \begin{align*}\frac{g^9 h^5}{6gh^{12}} \cdot \frac{18h^3}{g^8}\end{align*}
  11. \begin{align*}\frac{4a^{10} b^7}{12a^{-6}} \cdot \frac{9a^{-5} b^4}{20a^{11} b^{-8}}\end{align*}
  12. \begin{align*}\frac{-g^8 h}{6g^{-8}} \cdot \frac{9g^{15} h^9}{-h^{11}}\end{align*}
  13. Rewrite the following exponential pattern with positive exponents: \begin{align*}5^{-4}, 5^{-3}, 5^{-2}, 5^{-1}, 5^0, 5^1, 5^2, 5^3, 5^4\end{align*}.
  14. Evaluate each term in the pattern from #13.
  15. Fill in the blanks.

As the numbers increase, you ______________ the previous term by 5.

As the numbers decrease, you _____________ the previous term by 5.

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.2. 

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Negative Exponent Property

The negative exponent property states that \frac{1}{a^m} = a^{-m} and \frac{1}{a^{-m}} = a^m for a \neq 0.

Zero Exponent Property

The zero exponent property says that for all a \neq 0, a^0 = 1.

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