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

  1. Use exponent rules to explain why these expressions are equal to each other:
    • \(\left(5^{\frac13}\right)^2\)
    • \(\left(5^2\right)^{\frac13}\)
  2. Write \(5^{\frac23}\) using radicals.
  3. Write \(5^{\frac43}\) using radicals. Show your reasoning using exponent rules.

4.3: Fractional Powers Are Just Numbers

  1. 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)              
    1. Plot the points that you filled in.
      Horizontal axis, 0 to 2, by the fraction 1 over 3’s. Vertical axis, 0 to 4, by 0 point 2’s.
    2. Connect the points as smoothly as you can.

    3. 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.

  2. Let’s investigate \(2^{\frac13}\):

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

  3. Let’s investigate \(2^{\frac23}\):

    1. Write \(2^{\frac23}\) using radical notation.
    2. What is the value of \(\left( 2^{\frac23}\right)^3\)?
    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 \((1.26)^3 = 2.000376\).

  1. Explain why \(\sqrt[3]{2}\) is very close to \(1.26\). Is it larger or smaller than \(1.26\)?
  2. 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