# Exponents and Exponential Functions

## Objective

Define rational exponents and convert between rational exponents and roots.

## Common Core Standards

### Core Standards

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• N.RN.A.1 — Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)³ = 5(1/3)³ to hold, so (51/3)³ must equal 5.

• N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.

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• 8.EE.A.1

• 8.EE.A.2

• 8.NS.A.1

## Criteria for Success

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1. Understand a rational exponent as a base with a rational exponent as a power, including either a fraction or a decimal.
2. Write rational exponents using a radical, where the denominator of the fractional exponent defines the root and the numerator of the fractional exponent defines the power of the base.
3. Write radicals as exponential expressions with rational exponents.
4. Extend the properties of integer exponents to rational exponents.

## Anchor Problems

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### Problem 1

Below is an equation that is not true.

${{{{{10}0}^{1\over2}}}=50}$

a.   Why is the statement incorrect? What do you think the correct value of ${{{{10}0}^{1\over2}}}$ is?

b.   Consider the following pattern. Where does ${{{{10}0}^{1\over2}}}$ fit in?

${{{10}0}^3=1,000,000}$

${{{10}0}^2={10},000}$

${{{10}0}^1={{10}0}}$

${{{10}0}^0=1}$

c.   Consider rewriting the base ${{10}0}$ as a power of ${10}$. How does this shed light on the value of ${{{{10}0}^{1\over2}}}$?

${{{{10}0}^{1\over2}}}=(\square)^{1\over2}$

d.   Try out these other rational exponents:

${25^{1\over2}}$                    ${144^{1\over2}}$                   ${8^{1\over3}}$

#### Guiding Questions

• What do you think it means to have a fraction as an exponent?
• What are other values that ${{{{10}0}}^{1\over2}}$ is not? Why?
• How does the pattern of powers of ${{{10}0}}$ help you understand where a rational power would fit in? Based on the pattern, what do you think ${{{10}0}}^{3\over2}$ would be?
• How does writing ${{{10}0}}$ as a power of ${10}$ help you understand the value of ${{{{10}0}}^{1\over2}}$? What exponent rule can you use to simplify the expression?
• In general, what does the denominator of a fractional exponent indicate?

#### Notes

This problem is meant to engage students in conversation and experimentation around how to work with rational exponents. Give students some time with part (a) on their own or in small groups. You can then offer the strategies in parts (b) and (c) to move the discussion along if students have not discovered those strategies on their own. There is a good example of how this conversation happened in the blog author’s classroom in the link below.

#### References

Divisible By 3 Mistakes to the Half Power

Mistakes to the Half Power is made available by Andrew Stadel on Divisible by 3 under the CC BY-NC-SA 3.0 license. Accessed May 17, 2018, 2:54 p.m..

Modified by The Match Foundation, Inc.

### Problem 2

All of the following equations are true.

${\sqrt{x}=x^{1\over2}}$                ${\sqrt{x}=x^{1\over3}}$                ${(\sqrt{x})^2=x}$               ${x^{2\over3}=\sqrt{x^2}}$

Determine a general statement to represent the relationship between a radical and its exponential expression.

#### Guiding Questions

• What does the denominator of a fractional exponent represent?
• What does the numerator of a fractional exponent represent?
• Write ${8^{2\over3}}$ as a radical and then evaluate it.

### Problem 3

Write the radicals in exponential form and write the exponentials in radical form.

a.   ${5^{6\over5}}$

b.   ${4^{-{2\over3}}}$

c.   ${2n^{2\over5}}$

d.   ${\sqrt{7^2}}$

e.   ${1\over{\sqrt{5}}}$

f.   ${\sqrt{(3x)^5}}$

#### Guiding Questions

• How do you treat a negative exponent? Does this change when you have a rational exponent rather than an integer exponent?
• What is the base of the exponent ${{2\over5}}$ in part (c)?
• Compare and contrast parts (c) and (f).

## Problem Set

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The following resources include problems and activities aligned to the objective of the lesson. They can be used to create a problem set for class (for non-Fishtank Plus users), or as supplementary or additional resources to the pre-made Problem Set (for Fishtank Plus users).

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Henry explains why ${4^{3\over2}=8}$:

"I know that ${4^3}$ is ${{64}}$ and the square root of ${{64}}$ is $8$."

Here is Henrietta’s explanation for why ${4^{3\over2}=8}$:

"I know that ${\sqrt4=2}$ and the cube of $2$ is $8$. "

1. Are Henry and Henrietta correct? Explain.
2. Calculate $4^{5\over2}$ and $27^{2\over3}$ using Henry’s or Henrietta’s strategy.
3. Use both Henry and Henrietta’s reasoning to express ${x^{m\over n}}$ using radicals (here $m$ and $n$ are positive integers and we assume ${x>0}$).