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8.3: Zero, Negative, and Fractional Exponents

Difficulty Level: At Grade Created by: CK-12
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In the previous lessons, we have dealt with powers that are positive whole numbers. In this lesson, you will learn how to solve expressions when the exponent is zero, negative, or a fractional number.

Exponents of Zero: For all real numbers \begin{align*}\chi, \chi \neq 0, \chi^0=1\end{align*}.

Example: \begin{align*}\frac{\chi^4}{\chi^4} = \chi^{4-4} = \chi^0 = 1\end{align*}. This example is simplified using the Quotient of Powers Property.

Simplifying Expressions with Negative Exponents

The next objective is negative exponents. When we use the quotient rule and we subtract a greater number from a smaller number, the answer will become negative. The variable and the power will be moved to the denominator of a fraction. You will learn how to write this in an expression.

Example: \begin{align*}\frac{x^4}{x^6} =x^{4-6}=x^{-2}=\frac{1}{x^2}\end{align*}. Another way to look at this is \begin{align*}\frac{\chi \cdot \chi \cdot \chi \cdot \chi}{\chi \cdot \chi \cdot \chi \cdot \chi \cdot \chi \cdot \chi}\end{align*}. The four \begin{align*}\chi\end{align*}s on top will cancel out with four \begin{align*}\chi\end{align*}s on bottom. This will leave two \begin{align*}\chi\end{align*}s remaining on the bottom, which makes your answer look like \begin{align*}\frac{1}{\chi^2}\end{align*}.

Negative Power Rule for Exponents: \begin{align*}\frac{1}{\chi^n} = \chi^{-n}\end{align*} where \begin{align*}\chi \neq 0\end{align*}

Example: \begin{align*}\chi^{-6} \gamma^{-2}= \frac{1}{\chi^6} \cdot \frac{1}{\gamma^2} = \frac{1}{\chi^6 \gamma^2}\end{align*}. The negative power rule for exponents is applied to both variables separately in this example.

Multimedia Link: For more help with these types of exponents, watch this http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-241s.html - PH School video or visit the http://www.mathsisfun.com/algebra/negative-exponents.html - mathisfun website.

Example 1: Write the following expressions without fractions.

(a) \begin{align*}\frac{2}{x^2}\end{align*}

(b) \begin{align*}\frac{x^2}{y^3}\end{align*}


(a) \begin{align*}\frac{2}{x^2}=2x^{-2}\end{align*}

(b) \begin{align*}\frac{x^2}{y^3}=x^2y^{-3}\end{align*}

Notice in Example 1(a), the number 2 is in the numerator. This number is multiplied with \begin{align*}\chi^{-2}\end{align*}. It could also look like this, \begin{align*}2 \cdot \frac{1}{\chi^2}\end{align*} to be better understood.

Simplifying Expressions with Fractional Exponents

The next objective is to be able to use fractions as exponents in an expression.

Roots as Fractional Exponents: \begin{align*}\sqrt[m]{a^n}=a^{\frac{n}{m}}\end{align*}

Example: \begin{align*}\sqrt{a}=a^{\frac{1}{2}}, \sqrt[3]{a}=a^{\frac{1}{3}}, \sqrt[5]{a^2}=(a^2)^{\frac{1}{5}}=a^{\frac{2}{5}}=a^{\frac{2}{5}}\end{align*}

Example 2: Simplify the following expressions.

(a) \begin{align*}\sqrt[3]{\chi}\end{align*}

(b) \begin{align*}\sqrt[4]{\chi^3}\end{align*}


(a) \begin{align*}\chi^{\frac{1}{3}}\end{align*}

(b) \begin{align*}\chi^{\frac{3}{4}}\end{align*}

It is important when evaluating expressions that you remember the Order of Operations. Evaluate what is inside the parentheses, then evaluate the exponents, then perform multiplication/division from left to right, then perform addition/subtraction from left to right.

Example 3: Evaluate the following expression.

(a) \begin{align*}3 \cdot 5^2 - 10 \cdot 5+1\end{align*}

Solution: \begin{align*}3 \cdot 5^2-10 \cdot 5+1=3 \cdot 25-10 \cdot 5+1=75-50+1=26\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: Zero, Negative, and Fractional Exponents (14:04)

Simplify the following expressions. Be sure the final answer includes only positive exponents.

  1. \begin{align*}x^{-1} \cdot y^2\end{align*}
  2. \begin{align*}x^{-4}\end{align*}
  3. \begin{align*}\frac{x^{-3}}{x^{-7}}\end{align*}
  4. \begin{align*}\frac{1}{x}\end{align*}
  5. \begin{align*}\frac{2}{x^2}\end{align*}
  6. \begin{align*}\frac{x^2}{y^3}\end{align*}
  7. \begin{align*}\frac{3}{xy}\end{align*}
  8. \begin{align*}3x^{-3}\end{align*}
  9. \begin{align*}a^2b^{-3}c^{-1}\end{align*}
  10. \begin{align*}4x^{-1}y^3\end{align*}
  11. \begin{align*}\frac{2x^{-2}}{y^{-3}}\end{align*}
  12. \begin{align*}a^{\frac{1}{2}} \cdot a^{\frac{1}{3}}\end{align*}
  13. \begin{align*}\left(a^{\frac{1}{3}}\right)^2\end{align*}
  14. \begin{align*}\frac{a^{\frac{5}{2}}}{a^{\frac{1}{2}}}\end{align*}
  15. \begin{align*}\left(\frac{x^2}{y^3}\right)^{\frac{1}{3}}\end{align*}
  16. \begin{align*}\frac{x^{-3}y^{-5}}{z^{-7}}\end{align*}
  17. \begin{align*}(x^{\frac{1}{2}} y^{-\frac{2}{3}})(x^2 y^{\frac{1}{3}})\end{align*}
  18. \begin{align*}\left(\frac{a}{b}\right)^{-2}\end{align*}
  19. \begin{align*}(3a^{-2}b^2c^3)^3\end{align*}
  20. \begin{align*}x^{-3} \cdot x^3\end{align*}

Simplify the following expressions without any fractions in the answer.

  1. \begin{align*}\frac{a^{-3}(a^5)}{a^{-6}}\end{align*}
  2. \begin{align*}\frac{5x^6y^2}{x^8y}\end{align*}
  3. \begin{align*}\frac{(4ab^6)^3}{(ab)^5}\end{align*}
  4. \begin{align*}\left(\frac{3x}{y^{\frac{1}{3}}}\right)^3\end{align*}
  5. \begin{align*}\frac{4a^2b^3}{2a^5b}\end{align*}
  6. \begin{align*}\left(\frac{x}{3y^2}\right)^3 \cdot \frac{x^2y}{4}\end{align*}
  7. \begin{align*}\left(\frac{ab^{-2}}{b^3}\right)^2\end{align*}
  8. \begin{align*}\frac{x^{-3}y^2}{x^2y^{-2}}\end{align*}
  9. \begin{align*}\frac{3x^2y^{\frac{3}{2}}}{xy^{\frac{1}{2}}}\end{align*}
  10. \begin{align*}\frac{(3x^3)(4x^4)}{(2y)^2}\end{align*}
  11. \begin{align*}\frac{a^{-2}b^{-3}}{c^{-1}}\end{align*}
  12. \begin{align*}\frac{x^{\frac{1}{2}}y^{\frac{5}{2}}}{x^{\frac{3}{2}}y^{\frac{3}{2}}}\end{align*}

Evaluate the following expressions to a single number.

  1. \begin{align*}3^{-2}\end{align*}
  2. \begin{align*}(6.2)^0\end{align*}
  3. \begin{align*}8^{-4} \cdot 8^6\end{align*}
  4. \begin{align*}(16^{\frac{1}{2}})^3\end{align*}
  5. \begin{align*}5^0\end{align*}
  6. \begin{align*}7^2\end{align*}
  7. \begin{align*}\left(\frac{2}{3}\right)^3\end{align*}
  8. \begin{align*}3^{-3}\end{align*}
  9. \begin{align*}16^{\frac{1}{2}}\end{align*}
  10. \begin{align*}8^{\frac{-1}{3}}\end{align*}

In 43 – 45, evaluate the expression for \begin{align*}x=2, y=-1, z=3\end{align*}.

  1. \begin{align*}2x^2-3y^3+4z\end{align*}
  2. \begin{align*}(x^2-y^2)^2\end{align*}
  3. \begin{align*}\left(\frac{3x^2y^5}{4z}\right)^{-2}\end{align*}
  4. Evaluate \begin{align*}x^24x^3y^44y^2\end{align*} if \begin{align*}x=2\end{align*} and \begin{align*}y=-1\end{align*}.
  5. Evaluate \begin{align*}a^4(b^2)^3+2ab\end{align*} if \begin{align*}a=-2\end{align*} and \begin{align*}b=1\end{align*}.
  6. Evaluate \begin{align*}5x^2-2y^3+3z\end{align*} if \begin{align*}x=3, \ y=2,\end{align*} and \begin{align*}z=4\end{align*}.
  7. Evaluate \begin{align*}\left(\frac{a^2}{b^3}\right)^{-2}\end{align*} if \begin{align*}a=5\end{align*} and \begin{align*}b=3\end{align*}.
  8. Evaluate \begin{align*}3 \cdot 5^5 - 10 \cdot 5+1\end{align*}.
  9. Evaluate \begin{align*}\frac{2 \cdot 4^2-3 \cdot 5^2}{3^2}\end{align*}.
  10. Evaluate \begin{align*}\left(\frac{3^3}{2^2}\right)^{-2} \cdot \frac{3}{4}\end{align*}.

Mixed Review

  1. A quiz has ten questions: 7 true/false and 3 multiple choice. The multiple choice questions each have four options. How many ways can the test be answered?
  2. Simplify \begin{align*}3a^4 b^4 \cdot a^{-3} b^{-4}\end{align*}.
  3. Simplify \begin{align*}(x^4 y^2 \cdot xy^0)^5\end{align*}.
  4. Simplify \begin{align*}\frac{v^2}{-vu^{-2} \cdot u^{-1} v^4}\end{align*}.
  5. Solve for \begin{align*}n: -6(4n+3)=n+32\end{align*}.

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