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Fractional Exponents

Relate fractional exponents to nth roots

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Fractional Exponents

Suppose you want to find the cube root of a number, but your calculator only allows you to find square roots. However, it does allow you to find a number raised to any power, including non-integer powers. What would you do? Is there a way that you could rewrite the cube root of a number with a fractional exponent so that you could use your calculator to find the answer?

Fractional Exponents

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

You can write roots as fractional exponents using the rule \begin{align*}\sqrt[m]{a^n}=a^{\frac{n}{m}}\end{align*}anm=anm.

Let's simplify the following expressions by writing them without a root:

  1.  \begin{align*}\sqrt{a}\\a^{\frac{1}{2}} \end{align*}aa12
  2.  \begin{align*}\sqrt[3]{a}\\a^{\frac{1}{3}}\end{align*}a3a13 
  3.  \begin{align*}\sqrt[5]{a^2}\\(a^2)^{\frac{1}{5}}\\a^{\frac{2}{5}}\end{align*}a25(a2)15a25 
  4. \begin{align*}\sqrt[3]{\chi}\end{align*}χ3


  1.  \begin{align*}\sqrt[4]{\chi^3}\end{align*}χ34


Evaluating Expressions with Exponents

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, and then perform addition/subtraction from left to right.

Let's evaluate the following expression:

\begin{align*}3 \cdot 5^2 - 9^{1/2} \cdot 5+1\end{align*}35291/25+1

\begin{align*}3 \cdot 5^2-9^{1/2} \cdot 5+1=3 \cdot 25-\sqrt{9} \cdot 5+1=75-3\cdot 50+1=75-150+1=-74\end{align*}35291/25+1=32595+1=75350+1=75150+1=74





Example 1

Earlier, you were asked how you can write the cube root so that you can use your calculator to find cube root of a number. 

As shown in this concept, you can write a root as a fractional exponent where the denominator is the power of the root and the numerator is the power that is inside the root. 

A cube root can be written as the number to the \begin{align*}\frac{1}{3}\end{align*}13 power. 

\begin{align*}\sqrt[3]{x}= x^{\frac{1}{3}}\end{align*}x3=x13 

Example 2

Simplify \begin{align*}\left(\frac{3^2\cdot 36^{3/2}}{2^3}\right)^{2/5}\end{align*}(32363/223)2/5.

We start by simplifying using the fact that \begin{align*}36^{3/2}=(36^{1/2})^3\end{align*}363/2=(361/2)3.

\begin{align*}\left(\frac{3^2\cdot 36^{3/2}}{2^3}\right)^{2/5}= \left(\frac{3^2\cdot (36^{1/2})^3}{2^3}\right)^{2/5}= \left(\frac{3^2\cdot (\sqrt{36})^3}{2^3}\right)^{2/5} = \left(\frac{3^2\cdot 6^3}{2^3}\right)^{2/5}\end{align*}

Next we rearrange knowing that 6 and 2 have a common factor, 2.

\begin{align*}\left(\frac{3^2\cdot 6^3}{2^3}\right)^{2/5}=\left( 3^2 \cdot \left(\frac{6}{2}\right)^3\right)^{2/5} = \left( 3^2 \cdot 3^3\right)^{2/5} =\left(3^5\right)^{2/5} =3^{5\cdot 2/5}=3^2=9\end{align*}


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

  1. \begin{align*}\left(a^{\frac{1}{3}}\right)^2\end{align*}
  2. \begin{align*}\frac{a^{\frac{5}{2}}}{a^{\frac{1}{2}}}\end{align*}
  3. \begin{align*}\left(\frac{x^2}{y^3}\right)^{\frac{1}{3}}\end{align*}
  4. \begin{align*}\frac{x^{-3}y^{-5}}{z^{-7}}\end{align*}
  5. \begin{align*}(x^{\frac{1}{2}} y^{-\frac{2}{3}})(x^2 y^{\frac{1}{3}})\end{align*}

Simplify the following expressions without any fractions in the answer.

  1. \begin{align*}\left(\frac{3x}{y^{\frac{1}{3}}}\right)^3\end{align*}
  2. \begin{align*}\frac{4a^2b^3}{2a^5b}\end{align*}
  3. \begin{align*}\left(\frac{x}{3y^2}\right)^3 \cdot \frac{x^2y}{4}\end{align*}
  4. \begin{align*}\left(\frac{ab^{-2}}{b^3}\right)^2\end{align*}
  5. \begin{align*}\frac{x^{-3}y^2}{x^2y^{-2}}\end{align*}
  6. \begin{align*}\frac{3x^2y^{\frac{3}{2}}}{xy^{\frac{1}{2}}}\end{align*}
  7. \begin{align*}\frac{(3x^3)(4x^4)}{(2y)^2}\end{align*}
  8. \begin{align*}\frac{a^{-2}b^{-3}}{c^{-1}}\end{align*}
  9. \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*}(16^{\frac{1}{2}})^3\end{align*}
  2. \begin{align*}5^0\end{align*}
  3. \begin{align*}7^2\end{align*}
  4. \begin{align*}\left(\frac{2}{3}\right)^3\end{align*}
  5. \begin{align*}3^{-3}\end{align*}
  6. \begin{align*}16^{\frac{1}{2}}\end{align*}
  7. \begin{align*}8^{\frac{-1}{3}}\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*}.

Review (Answers)

To view the Review answers, open this PDF file and look for section 8.4. 

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