# 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.)

 $$x$$ $$2^x$$ (using exponents) $$2^x$$ (decimal approximation) -2 $$\text{-}\frac53$$ $$\text{-}\frac43$$ -1 $$\text{-}\frac23$$ $$\text{-}\frac13$$ 0 $$2^{\text- 2}$$ $$2^{\text{-}\frac53}$$ $$2^{\text{-}\frac43}$$ $$2^{\text- 1}$$ $$2^{\text{-}\frac23}$$ $$2^{\text{-}\frac13}$$ $$2^0$$
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.
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. $$8^{\frac45}$$, $$\sqrt{8}^5$$, $$\sqrt{8}^4$$
2. $$8^{\text{-} \frac45}$$, $$\dfrac{1}{\sqrt{8^4}}$$, $$\text-\dfrac{1}{\sqrt{8^4}}$$
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. $$\dfrac{1}{\left(\sqrt{5}\right)^6}$$

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.

 $$x$$ $$5^x$$ (using exponents) $$5^x$$ (equivalent expressions) -1 $$\text{-} \frac23$$ $$\text{-} \frac13$$ 0 $$\frac13$$ $$\frac23$$ 1 $$5^{\text-1}$$ $$5^{\text{-} \frac23}$$ $$5^{\text{-} \frac13}$$ $$5^0$$ $$5^{\frac13}$$ $$5^{\frac23}$$ $$5^1$$ $$\frac15$$ $$\dfrac{1}{\sqrt{5^2}}$$ or $$\dfrac{1}{\sqrt{25}}$$ $$\dfrac{1}{\sqrt{5}}$$ 1 $$\sqrt{5}$$ $$\sqrt{5^2}$$ or $$\sqrt{25}$$ 5