Lesson 5
Negative Rational Exponents
 Let’s investigate negative exponents.
5.1: Math Talk: Don’t Be Negative
Evaluate mentally.
\(9^2\)
\(9^{\text2}\)
\(9^{\frac12}\)
\(9^{\text\frac12}\)
5.2: Negative Fractional Powers Are Just Numbers

Complete the table as much as you can without using a calculator. (You should be able to fill in three spaces.)
\(x\) 2 \(\text{}\frac53\) \(\text{}\frac43\) 1 \(\text{}\frac23\) \(\text{}\frac13\) 0 \(2^x\) (using exponents) \(2^{\text 2}\) \(2^{\text{}\frac53}\) \(2^{\text{}\frac43}\) \(2^{\text 1}\) \(2^{\text{}\frac23}\) \(2^{\text{}\frac13}\) \(2^0\) \(2^x\) (decimal approximation)  Plot these powers of 2 in the coordinate plane.
 Connect the points as smoothly as you can.
 Use your graph of \(y=2^x\) to estimate the value of the other powers in the table, and write your estimates in the table.

Let’s investigate \(2^{\text{} \frac13}\).
 Write \(2^{\text{} \frac13}\) using radical notation.
 What is the value of \(\left( 2^{\text{} \frac13}\right)^3\)?
 Raise your estimate of \(2^{\text{} \frac13}\) to the third power. What should it be? How close did you get?

Let’s investigate \(2^{\text{} \frac23}\).
 Write \(2^{\text{} \frac23}\) using radical notation.
 What is \(\left( 2^{\text{} \frac23}\right)^3\)?
 Raise your estimate of \(2^{\text{} \frac23}\) to the third power. What should it be? How close did you get?
5.3: Any Fraction Can Be an Exponent
 For each set of 3 numbers, cross out the expression that is not equal to the other two expressions.
 \(8^{\frac45}\), \(\sqrt[4]{8}^5\), \(\sqrt[5]{8}^4\)
 \(8^{\text{} \frac45}\), \(\dfrac{1}{\sqrt[5]{8^4}}\), \(\text\dfrac{1}{\sqrt[5]{8^4}}\)
 \(\sqrt{4^3}\), \(4^{\frac32}\), \(4^{\frac23}\)
 \(\dfrac{1}{\sqrt{4^3}}\), \(\text4^{\frac32}\), \(4^{\text\frac32}\)
 For each expression, write an equivalent expression using radicals.
 \(17^{\frac32}\)
 \(31^{\text{} \frac32}\)
 For each expression, write an equivalent expression using only exponents.
 \(\left(\sqrt{3}\right)^4\)
 \(\dfrac{1}{\left(\sqrt[3]{5}\right)^6}\)
Write two different expressions that involve only roots and powers of 2 which are equivalent to \(\frac{4^\frac23}{8^\frac14}\).
5.4: Make These Exponents Less Complicated
Match expressions into groups according to whether they are equal. Be prepared to explain your reasoning.
\(\left(\sqrt{3}\right)^4\)
\(\sqrt{3^2}\)
\(\left (3^{\frac12}\right )^4\)
\((\sqrt{3})^2 \boldcdot (\sqrt{3})^2\)
\(\left (3^2 \right) ^{\frac12}\)
\(3^2\)
\(3^{\frac42}\)
\(\left (3^{\frac12}\right)^2\)
Summary
When we have a number with a negative exponent, it just means we need to find the reciprocal of the number with the exponent that has the same magnitude, but is positive. Here are two examples:
\(\displaystyle 7^{\text{} 5} = \dfrac{1}{7^5}\)
\(\displaystyle 7^{\text{} \frac65} = \dfrac{1}{7^{\frac65}}\)
The table shows a few more examples of exponents that are fractions and their radical equivalents.
\(x\)  1  \(\text{} \frac23\)  \(\text{} \frac13\)  0  \(\frac13\)  \(\frac23\)  1 

\(5^x\) (using exponents)  \(5^{\text1}\)  \(5^{\text{} \frac23}\)  \(5^{\text{} \frac13}\)  \(5^0\)  \(5^{\frac13}\)  \(5^{\frac23}\)  \(5^1\) 
\(5^x\) (equivalent expressions)  \(\frac15\)  \(\dfrac{1}{\sqrt[3]{5^2}}\) or \(\dfrac{1}{\sqrt[3]{25}}\)  \(\dfrac{1}{\sqrt[3]{5}}\)  1  \(\sqrt[3]{5}\)  \(\sqrt[3]{5^2}\) or \(\sqrt[3]{25}\)  5 