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Negative Exponents

Any value to the zero power equals 1, convert negative exponents

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Negative Exponents

All the students in a class were randomly given an expression, and they were asked to make pairs, with one boy and one girl per pair. The students were asked to divide the boy's expression by the girl's expression. Bill and Jenna paired up, with Bill having \begin{align*}10x^4\end{align*} and Jenna having \begin{align*}5x^5\end{align*}. Also, Tim and Meg paired up, with Tim having \begin{align*}7y^3\end{align*}and Meg having \begin{align*}14y^3\end{align*}. What is Bill and Jenna's quotient? How about Tim and Meg's quotient?

Zero Power and Negative Exponents

Previously, we have dealt with powers that are positive whole numbers. Now, you will learn how to solve expressions when the exponent is zero or a negative number.

For all real numbers \begin{align*}\chi, \chi \neq 0, \chi^0=1\end{align*}. This means that any number raised to the 0th power is 1.

Let's simplify \begin{align*}\frac{\chi^4}{\chi^4}\end{align*}:

\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.

Let's simplify \begin{align*}\frac{x^4}{x^6}\end{align*}:

\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 the 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*}.

If you want to simplify this expression any further, you need to use the Negative Power Rule for Exponents.

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

Now, let's rewrite the expression  \begin{align*}\chi^{-6} \gamma^{-2}\end{align*} using only positive exponents:

\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.

Finally, let's write the following expressions without fractions:

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


  1. \begin{align*}\frac{x^2}{y^3}\end{align*}


Notice that in question 1, the number 2 is in the numerator. This number is multiplied with \begin{align*}x^{-2}\end{align*}. It could also look like \begin{align*}2 \cdot \frac{1}{x^2}\end{align*} to be better understood.





Example 1

Earlier, you were told about a class where students had to make pairs based on the expressions that they were randomly given. The pairs had to be one girl and one boy and they were required to divide the boy's expression by the girl's expression. In the pair of Bill and Jenna, Bill is given \begin{align*}10x^4\end{align*} and Jenna is given \begin{align*}5x^5\end{align*}. In the pair of Tim and Meg, Tim has \begin{align*}7y^3\end{align*} and Meg has \begin{align*}14y^3\end{align*}. What is Bill and Jenna's quotient? What is Tim and Meg's quotient?

Bill and Jenna's quotient is:

\begin{align*}\frac{10x^4}{5x^5}&= 2x^{4-5}\\ &=2x^{-1}\\ &=\frac{2}{x}\end{align*} 

Tim and Meg's quotient is:

\begin{align*}\frac{7y^3}{14y^3}&=\frac{1}{2}\cdot y^{3-3}\\ &=\frac{1}{2}\cdot y^0\\ &= \frac{1}{2}\end{align*}

Example 2

Simplify \begin{align*}\left( \frac{x^2y^{-3}}{x^5y^2}\right)^{2}\end{align*}, giving the answer with only positive exponents.

\begin{align*}\left( \frac{x^2 y^{-3}}{x^5 y^2} \right)^2 = \left(x^2 x^{-5} y^{-3} y^{-2}\right)^2=\left(x^{2-5} y^{-3-2}\right)^2=(x^{-3} y^{-5})^2= (x^{-3})^2 (y^{-5})^2 &= x^{(-3)(2)} y^{(-5)(2)} = x^{-6} y^{-10}=\frac{1}{x^6y^{10}} \end{align*}


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*}\left(\frac{a}{b}\right)^{-2}\end{align*}
  13. \begin{align*}(3a^{-2}b^2c^3)^3\end{align*}
  14. \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*}

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*}

In 21 – 23, evaluate the expression for \begin{align*}x=2, y=-1, \text{and } 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*}.

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.3. 

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exponents of zero

For all real numbers \chi, \chi \neq 0, \chi^0=1.

Negative Power Rule for Exponents

\frac{1}{\chi^n} = \chi^{-n} where \chi \neq 0.

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.

quotient rule

In calculus, the quotient rule states that if f and g are differentiable functions at x and g(x) \ne 0, then \frac {d}{dx}\left [ \frac{f(x)}{g(x)} \right ]= \frac {g(x) \frac {d}{dx}\left [{f(x)} \right ] - f(x) \frac{d}{dx} \left [{g(x)} \right ]}{\left [{g(x)} \right ]^2}.

Zero Exponent Property

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

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