Exponents and Exponential Functions

Lesson 8


Simplify radical expressions.

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. Rewrite radical expressions by evaluating any possible roots from inside the radical. For example, simplify a square root so that no perfect squares remain inside the square root. 
  2. Understand that $${\sqrt[n]{a^n}=a}$$.

Tips for Teachers


Simplifying radicals will be a useful skill for when students compute with radicals in the upcoming lessons. It is helpful for students to be familiar with and to have readily in mind the perfect squares and cubes within 100.

Anchor Problems


Problem 1

Determine which value is greater without calculating the value of the radical expression.

a.   $${11\sqrt3}$$ _____ $${13\sqrt3}$$

b.   $${\sqrt{72}}$$ _____ $${5\sqrt2}$$

c.   $${\sqrt{75}}$$ _____ $${2\sqrt{27}}$$

Guiding Questions

  • Why is it difficult to compare the radical expressions as they are written? What if each pair had the same value under the radical? Would you be able to more easily compare them?
  • Are there any perfect square factors in $${72}$$? How can you rewrite $$\sqrt{{72}}$$ with a perfect square as a factor? What does the perfect square factor evaluate to?
  • Are there any perfect square factors in $${75}$$ or $${27}$$?


If students have a difficult time identifying the perfect squares right away, they can use prime factorization to break down the numbers as prime factors. For example, $${\sqrt{75}=\sqrt{3\cdot5\cdot5}}$$, and $${\sqrt{5\cdot5}=5}$$, therefore $${\sqrt{75}=5\sqrt3}$$.

Problem 2

Rewrite each expression as a radical with no perfect squares or cubes remaining inside the radical.

a.   $${\sqrt{32x^5y^2}}$$

b.   $${2\sqrt[3]{540m^7n^5}}$$

c.   $${(24x^2)^{1\over2}}$$

d.   $${(24x^2)^{1\over3}}$$

e.   $${(24x^2)^{2\over3}}$$

Guiding Questions

  • How is simplifying a cube root similar to and different from simplifying a square root?
  • How is simplifying a variable under a radical similar to and different from simplifying a numerical value?
  • What properties of exponents are you using?

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


Simplify each radical so there are no perfect squares or cubes remaining inside the radical.

a.   $${\sqrt{54x^8y^5}}$$

b.   $${\sqrt[3]{54x^8y^5}}$$