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

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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

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Problem 3

Compute and simplify.

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

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

Guiding Questions

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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.

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