Lesson 4
Positive Rational Exponents
 Let’s use roots to write exponents that are fractions.
4.1: Math Talk: Regrouping Fractions
Find the value of each expression mentally.
\(\frac12 \boldcdot 5 \boldcdot 4\)
\(\frac52 \boldcdot 4\)
\(\frac23 \boldcdot 7 \boldcdot \frac32\)
\(7 \boldcdot \frac53 \boldcdot \frac37\)
4.2: You Can Use Any Fraction As an Exponent
 Use exponent rules to explain why these expressions are equal to each other:
 \(\left(5^{\frac13}\right)^2\)
 \(\left(5^2\right)^{\frac13}\)
 Write \(5^{\frac23}\) using radicals.
 Write \(5^{\frac43}\) using radicals. Show your reasoning using exponent rules.
4.3: 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.)
\(x\) 0 \(\frac13\) \(\frac23\) 1 \(\frac43\) \(\frac53\) 2 \(2^x\) (using exponents) \(2^0\) \(2^{\frac13}\) \(2^{\frac23}\) \(2^1\) \(2^{\frac43}\) \(2^{\frac53}\) \(2^2\) \(2^x\) (decimal approximation)  Plot the points that you filled in.
 Connect the points as smoothly as you can.

Use this graph of \(y=2^x\) to estimate the value of the other powers in the table, and write your estimates in the table.
 Plot the points that you filled in.

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

Let’s investigate \(2^{\frac23}\):
 Write \(2^{\frac23}\) using radical notation.
 What is the value of \(\left( 2^{\frac23}\right)^3\)?
 Raise your estimate from the table of \(2^{\frac23}\) to the third power. What should it be? How close did you get?
Answer these questions using the fact that \((1.26)^3 = 2.000376\).
 Explain why \(\sqrt[3]{2}\) is very close to \(1.26\). Is it larger or smaller than \(1.26\)?
 Is it possible to write \(\sqrt[3]{2}\) exactly with a finite decimal expansion? Explain how you know.
Summary
Using exponent rules, we know \(3^{\frac14}\) is the same as \(\sqrt[4]{3}\) because \(\left(3^{\frac14}\right)^4 =3\). But what about \(3^{\frac54}\)?
Using exponent rules,
\(\displaystyle 3^{\frac54}=\left(3^5\right)^{\frac14}\)
which means that
\(\displaystyle 3^{\frac54}=\sqrt[4]{3^5}\)
Since \(3^5=243\), we could just write \(3^{\frac54}=\sqrt[4]{243} \).
Alternatively, we could express the fraction \(\frac54\) as \(\frac14 \boldcdot 5\) instead. Using exponent rules, we get
\(\displaystyle 3^{\frac54}= \left(3^{\frac14}\right)^5= \left(\sqrt[4]{3}\right)^5\)
Here are more examples of exponents that are fractions and their equivalents:
\(x\)  0  \(\frac13\)  \(\frac23\)  1  \(\frac43\)  \(\frac53\)  2 

\(5^x\) (using exponents)  \(5^0\)  \(5^{\frac13}\)  \(5^{\frac23}\)  \(5^1\)  \(5^{\frac43}\)  \(5^{\frac53}\)  \(5^2\) 
\(5^x\) (equivalent expression)  1  \(\sqrt[3]{5}\)  \(\sqrt[3]{5^2}\) or \(\sqrt[3]{25}\)  5  \(\sqrt[3]{5^4}\) or \(\sqrt[3]{625}\)  \(\sqrt[3]{5^5}\) or \(\sqrt[3]{3125}\)  25 