Extracting Roots and Rewriting Radicals - HHS Algebra II

[Pages:44]Keepin' It Real

10.4

Extracting Roots and Rewriting Radicals

LEARNING GOALS

In this lesson, you will:

? Extract roots from radicals. ? Rewrite radicals as powers that have rational exponents. ? Rewrite powers that have rational exponents as radicals.

? Carnegie Learning

Radicals can p_ro__duce imaginary results. For example, the square root of 24 is equal to 2i, 24 5 2i. But, in this chapter we are not going to talk about imaginary numbers. We are going to keep it real!

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PROBLEM 1 Root of the Problem

Previously, you have rewritten radicals by extracting roots involving numbers. In this lesson you will explore how to extract roots for expressions of the form n xn. To determine how to extract a variable from a radical, let's consider several different values of n.

1. For each value of n for the expression n xn, complete the table and sketch the graph. Then identify the function family associated with the graph and write the corresponding equation.

a. Let n 5 2.

x

xn 5 x2

n xn5 2 x2

y 8

22

6

4

21

2

0

28 26 24 22 0 2 4 6 8 x

1

22

2

24

26

28

Function family of the graph: Equation of the graph:

b. Let n 5 3.

x

xn 5 x3

22

21

0

1

2

n xn5 3 x3

Function family of the graph: Equation of the graph:

y 8 6 4 2

28 26 24 22 0 22 24 26 28

2 4 6 8x

? Carnegie Learning

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Chapter 10 Radical Functions

c. Let n 5 4.

x

xn 5 x4

22

21

0

1

2

n xn5 4 x4

Function family of the graph: Equation of the graph:

y 8 6 4 2

28 26 24 22 0 22 24 26 28

2 4 6 8x

d. Let n 5 5.

x

xn 5 x5

22

21

0

1

2

n xn5 5 x5

Function family of the graph: Equation of the graph:

y 8 6 4 2

28 26 24 22 0 22 24 26 28

2 4 6 8x

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10.4 Extracting Roots and Rewriting Radicals

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e. Analyze your representations for each value of n. What do you notice?

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To extract a variable from a radical, the expression n xncan be written as:

n xn5

|x|, x,

when n is even when n is odd

2. Explain why 7 x75 |x| is incorrect, for real values of x.

One way to say 7 x7is "the seventh ro t of x to the seventh."

? __ 3. Asia and Melissa shared their work for extracting the root from x4, for real values of x.

Asia x45 |x2|

Melissa_x_4 5 x2

Who's correct? Explain your reasoning.

? Carnegie Learning

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Chapter 10 Radical Functions

PROBLEM 2 Sort It Out

Let's review the properties of powers. 1. Write an explanation for each property to complete the table.

Property of Powers

Rule

Written Explanation

Product of Powers

am ? an 5 am 1 n

Quotient of Powers

_a_m_ a n

5

a m

2

n

Power to a Power

(am)n 5 amn

Product to a Power

(am ? bn)p 5 amp ? bnp

Quotient to a Power

( )_a_m_ p 5 _a_m_p

b n

bnp

Zero Power

a0 5 1, if a fi 0

Negative Exponent In Numerator

a2m 5 _a1_m_, if a fi 0 and m . 0

Negative Exponent In Denominator

__1__ a2m

5

a m ,

if a fi 0 and m . 0

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? Carnegie Learning

10.4 Extracting Roots and Rewriting Radicals

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? Carnegie Learning

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Chapter 10 Radical Functions

2. Cut out the items and tape each item into the appropriate group on the next page.

a 0 (2a)4 a3 ? a a0 ? a4 a ? a25

_a _ 6 a 6

_a _ 3 a 7

_a _ 6 a 2

( )_1_ 2

a 2

( )a_1_ 8 2

a24 ? a0

(a212)_31_

a4 ? a24 ( a 2 ) 2 (ab2)2 (a8b4)_21_

( )_b_1_2

_1_ 3

a26

( )_b__ 2

a22

_1_ a 4

_1__ a24

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? Carnegie Learning

10.4 Extracting Roots and Rewriting Radicals

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? Carnegie Learning

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Chapter 10 Radical Functions

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