Exponents and Exponential Functions

Lesson 9

Objective

Multiply and divide rational exponent expressions and radical expressions.

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.

  • N.RN.B.3 — Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Foundational Standards

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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 that $${{\sqrt[n]{a}}\times\sqrt[n]{b}=\sqrt[n]{ab}}$$, and that $${{{{\sqrt[n]{a}}}\over{\sqrt[n]{b}}}=\sqrt[n]{a\over b}}$$.
  2. Understand that $${\sqrt[n]{a}}$$ and $${\sqrt[m]{a}}$$ can be rewritten as $${a^{1\over n}}$$ and $${a^{1\over m}}$$ in order to be multiplied or divided. 
  3. Apply the properties of exponents and properties of operations to multiply and divide rational exponent and radical expressions.
  4. Understand that a rational number multiplied by a rational number is rational, and a rational number multiplied by an irrational number is irrational. 

Anchor Problems

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Problem 1

All of the following equations are true.

$${\sqrt{16}\cdot\sqrt9=\sqrt{144}}$$              $${{\sqrt{80}\over\sqrt{4}}=\sqrt{20}}$$               $${\sqrt5\cdot\sqrt[3]{5}=\sqrt[6]{5^5}}$$

a.   What general rules can you determine from these examples?

b.   Find the products or quotients below

i.   $${\sqrt{12}\cdot\sqrt2}$$

ii.  $${\sqrt[3]{4}\cdot\sqrt[3]{3}\cdot\sqrt[3]{5}}$$

iii.   $${\sqrt2\cdot\sqrt3\cdot\sqrt[3]{6}}$$

iv.   $${\sqrt[3]{45}\over\sqrt[3]{5}}$$

Guiding Questions

  • What do you notice happens when you multiply two square roots? 
  • What do you notice happens when you divide two square roots?
  • How can you express these occurrences as a general rule?
  • What if the radicals have different indices (e.g., square root and cube root)? Can you multiply the numbers under the radicals (the radicands)?
  • When is it useful to change a radical expression into a rational exponent expression?

Problem 2

Multiply and simplify as much as possible.

a.   $${-5\sqrt{12}\cdot\sqrt8}$$

 

b.   $${\sqrt[3]{6x^2}\cdot\sqrt[3]{9x^4}}$$

Divide and simplify as much as possible.

c.   $${2\sqrt{6}\div\sqrt{24}}$$

d.   $${\sqrt{120m^9}\over{10\sqrt{4m^4}}}$$

Guiding Questions

  • When is it the best approach to multiply the radicands (expressions under the radicals)?
  • When is it the best approach to change the radicals into rational exponent expressions? 
  • Can you simplify the expression $${\sqrt{5}\over5}$$? Why or why not? Can you simplify the expression $${\sqrt{5}\over\sqrt5}$$? Why or why not? 
  • Are either of your factors irrational? Is your product irrational?
  • Is either the divisor or dividend irrational? Is your quotient irrational?

Problem 3

Compute and simplify.

a.   $${\sqrt{10}\cdot2\sqrt[3]{10}}$$

b.   $${3\sqrt[3]{16}\div2\sqrt[3]{54}}$$

Guiding Questions

  • When is it the best approach to multiply the radicands (expressions under the radicals)?
  • When is it the best approach to change the radicals into rational exponent expressions? 
  • Are either of your factors irrational? Is your product irrational?
  • Is either the divisor or dividend irrational? Is your quotient irrational?

Notes

This Anchor Problem is optional and can be moved to the problem set depending on the timing of your class and/or needs of your students.

Problem Set

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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

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Find the error in each solution. Then find the correct product or quotient.

a.   $${\sqrt{20}\cdot\sqrt[3]{5}=\sqrt{100}=10}$$

b.   $${{{4\sqrt{35}}\over{\sqrt{28}}}={{4\sqrt{5\cdot7}}\over{\sqrt{4\cdot7}}}={\sqrt5}}$$