# Exponents and Exponential Functions

## Objective

Write equivalent radical and rational exponent expressions. Identify quantities as rational or irrational.

## Common Core Standards

### Core Standards

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• 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. Convert between radicals and rational exponents fluently.
2. Apply the properties of exponents to convert between radicals and rational exponents.
3. Determine if a given number or expression is rational or irrational.

## Anchor Problems

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### Problem 1

For the problems below, determine whether each equation is True or False.

 Expression True? False? a.   ${\sqrt{32}=2^{5\over2}}$ b.   ${16^{3\over2}=8^2}$ c.   ${4^{1\over2}=\sqrt{64}}$ d.   ${2^8=(\sqrt{16})^6}$ e.   ${(\sqrt{64})^{1\over3}=8^{1\over6}}$

#### References

Smarter Balanced Assessment Consortium: Item and Task Specifications MAT.HS.SR.1.00NRN.A.152

MAT.HS.SR.1.00NRN.A.152 from Development and Design: Item and Task Specifications made available by Smarter Balanced Assessment Consortium.  © The Regents of the University of California – Smarter Balanced Assessment Consortium. Accessed May 17, 2018, 11:29 a.m..

### Problem 2

In each of the following problems, a number is given. If possible, determine whether the given number is rational or irrational. In some cases, it may be impossible to determine whether the given number is rational or irrational. Justify your answers.

a.   ${4+\sqrt7}$

b.   ${\sqrt{45}\over\sqrt{5}}$

c.   ${6\over \pi}$

d.   ${\sqrt2 + \sqrt3}$

e.   ${{2+\sqrt{7}}\over{2a+\sqrt{7a^2}}}$, where $a$ is a positive integer

f.   ${x+y}$, where $x$ and $y$ are irrational numbers

#### References

Illustrative Mathematics Rational or Irrational?

Rational or Irrational?, accessed on May 17, 2018, 11:34 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

## 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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Provide a written explanation for each question below.

a.   Is it true that ${\left(1000^{1\over3}\right)^3=(1000^3)^{1\over3}}$? Explain or show how you know.

b.   Is it true that ${\left(4^{1\over2}\right)^3=(4^3)^{1\over2}}$? Explain or show how you know.

c.   Suppose that $m$ and $n$ are positive integers and $b$ is a real number so that the principal $n^{th}$ root of $b$ exists. In general, does $\left(b^{1\over n}\right)^m=(b^m)^{1\over n}$? Explain or show how you know.

#### References

EngageNY Mathematics Algebra II > Module 3 > Topic A > Lesson 3Exit Ticket, Question #3

Algebra II > Module 3 > Topic A > Lesson 3 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.