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.)

 $$x$$ $$2^x$$ (using exponents) $$2^x$$ (decimal approximation) 0 $$\frac13$$ $$\frac23$$ 1 $$\frac43$$ $$\frac53$$ 2 $$2^0$$ $$2^{\frac13}$$ $$2^{\frac23}$$ $$2^1$$ $$2^{\frac43}$$ $$2^{\frac53}$$ $$2^2$$
1. Plot the points that you filled in.
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:

 $$x$$ $$5^x$$ (using exponents) $$5^x$$ (equivalent expression) 0 $$\frac13$$ $$\frac23$$ 1 $$\frac43$$ $$\frac53$$ 2 $$5^0$$ $$5^{\frac13}$$ $$5^{\frac23}$$ $$5^1$$ $$5^{\frac43}$$ $$5^{\frac53}$$ $$5^2$$ 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