Exponents and Exponential Functions

Lesson 7


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

Common Core Standards

Core Standards


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

Foundational Standards


  • 8.EE.A.1

  • 8.EE.A.2

  • 8.NS.A.1

Criteria for Success


  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


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.   $${4^{1\over2}=\sqrt[4]{64}}$$



d.   $${2^8=(\sqrt[3]{16})^6}$$



e.   $${(\sqrt{64})^{1\over3}=8^{1\over6}}$$


Guiding Questions

  • How can you rewrite some of the bases so they are the same within each problem? 
  • What properties of exponents are useful to use to show equivalence?
  • How else can you check that each equation is true or false besides using properties of exponents?
  • Which expressions give an irrational value?
  • Which expressions give a rational value? 


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

Guiding Questions

  • Recall, what is an irrational number? How is it different from a rational number?
  • Give some examples of irrational numbers and explain what makes them irrational.
  • If you add an irrational number to another irrational number, do you think you’ll get an irrational sum? Why or why not?


Students have not yet studied how to simplify radicals or how to multiply or divide with them. Part (b) may be challenging for students to answer at this point; however, it is worthwhile for them to try it out and reason through an explanation.


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


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).

Target Task


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.