Write equivalent radical and rational exponent expressions. Identify quantities as rational or irrational.
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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. $${4^{1\over2}=\sqrt[4]{64}}$$ 


d. $${2^8=(\sqrt[3]{16})^6}$$ 


e. $${(\sqrt{64})^{1\over3}=8^{1\over6}}$$ 
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..
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
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 BYNCSA 4.0. For further information, contact Illustrative Mathematics.
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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.
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 BYNCSA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..
Modified by The Match Foundation, Inc.