# Exponents and Exponential Functions

## 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.

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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 ${x\sqrt[n]{a}+y\sqrt[n]{a}=(x+y)\sqrt[n]{a}}$.
2. Understand that radicals with different indices or radicands cannot be combined.
3. Apply the properties of exponents and properties of operations to add and subtract rational exponent and radical expressions.
4. Understand that a rational number added to or subtracted from a rational number is rational, and a rational number added to or subtracted from an irrational number is irrational.

## Anchor Problems

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### Problem 1

The statement below is incorrect. Explain why it is incorrect and then correct the statement.

${\sqrt9+\sqrt9=\sqrt{18}}$

Which statements below are true? Select all that apply.

a.   ${2\sqrt3+\sqrt3=3\sqrt3}$

b.   ${10\sqrt5-5\sqrt5=5}$

c.   ${\sqrt2+\sqrt{2}=2\sqrt2}$

d.   ${4\sqrt8-3\sqrt8=\sqrt8}$

e.   ${5\sqrt{8}-2\sqrt{6}=3\sqrt2}$

f.   ${8\sqrt[n]{x}-6\sqrt[n]{x}=2\sqrt[n]{x}}$

#### Guiding Questions

• When adding and subtracting radicals, how are the radical terms treated similarly to “like terms” in algebraic expressions?
• Can you combine two radicals if they have different indices (e.g., can you combine a square root and a cube root)?
• What are the conditions under which you can add or subtract radicals?
• How is this similar to or different from multiplying and dividing radicals?

### Problem 2

Compute and simplify.

a.   ${\sqrt{48}+2\sqrt{27}-3\sqrt{12}}$

b.   ${-5\sqrt{20}-\sqrt{72}+\sqrt{125}}$

c.   ${(\sqrt{16}+\sqrt{54})^3}$

#### Guiding Questions

• At first look, none of the radicands are the same in the problems above. Can you simplify the radicals so they can be combined?
• What is your strategy for part (c)?

## 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).

a.   ${\sqrt{8a}-\sqrt{32a}}$
b.   ${8\sqrt{18}-4\sqrt{8}-\sqrt{24}}$
c.   ${\sqrt{3}(2\sqrt{18}+\sqrt{32})}$