Lesson 5

Negative Rational Exponents

  • Let’s investigate negative exponents.

5.1: Math Talk: Don’t Be Negative

Evaluate mentally.

\(9^2\)

\(9^{\text-2}\)

\(9^{\frac12}\)

\(9^{\text-\frac12}\)

5.2: Negative 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.)

    \(x\) -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)
    1. Plot these powers of 2 in the coordinate plane. ​​​​​​
    2. Connect the points as smoothly as you can.
    3. 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.
    Horizontal axis, -2 to 0, by the fraction 1 over 3’s. Vertical axis, 0 to 1, by 0.2’s.

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

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

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

    1. Write \(2^{\text{-} \frac23}\) using radical notation.
    2. What is \(\left( 2^{\text{-} \frac23}\right)^3\)?
    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

  1. For each set of 3 numbers, cross out the expression that is not equal to the other two expressions.
    1. \(8^{\frac45}\), \(\sqrt[4]{8}^5\), \(\sqrt[5]{8}^4\)
    2. \(8^{\text{-} \frac45}\), \(\dfrac{1}{\sqrt[5]{8^4}}\), \(\text-\dfrac{1}{\sqrt[5]{8^4}}\)
    3. \(\sqrt{4^3}\), \(4^{\frac32}\), \(4^{\frac23}\)
    4. \(\dfrac{1}{\sqrt{4^3}}\), \(\text-4^{\frac32}\), \(4^{\text-\frac32}\)
  2. For each expression, write an equivalent expression using radicals.
    1. \(17^{\frac32}\)
    2. \(31^{\text{-} \frac32}\)
  3. For each expression, write an equivalent expression using only exponents.
    1. \(\left(\sqrt{3}\right)^4\)
    2. \(\dfrac{1}{\left(\sqrt[3]{5}\right)^6}\)


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) \(5^{\text-1}\) \(5^{\text{-} \frac23}\) \(5^{\text{-} \frac13}\) \(5^0\) \(5^{\frac13}\) \(5^{\frac23}\) \(5^1\)
\(5^x\) (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