# 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:

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

#### 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, 10:54 a.m..

Modified by Fishtank Learning, Inc.

### Problem 2

All of the following equations are true.

${\sqrt{x}=x^{1\over2}}$                ${\sqrt[3]{x}=x^{1\over3}}$                ${(\sqrt{x})^2=x}$               ${x^{2\over3}=\sqrt[3]{x^2}}$

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

### 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.   ${\sqrt[3]{7^2}}$

e.   ${1\over{\sqrt[3]{5}}}$

f.   ${\sqrt{(3x)^5}}$

## Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

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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 ${x>0}$).