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Pearson Algebra 2 Common Core, 2011

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Pearson Algebra 2 Common Core, 2011 View details

2. Multiplying and Dividing Radical Expressions

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Exercise 86 Page 373

If $n$ is an odd number, then the radical expression $n a^{n}$ simplifies to $a .$ If $n$ is even, the expression $n a^{n}$ simplifies to $∣ a ∣ .$

$9 c^{24} d^{32}$

Practice makes perfect

For any real number

a,

the radical expression

n a^{n}

can be simplified as follows.

n a^{n} = {∣ a ∣ if n is odd ∣ a ∣ if n is even

Since the radical is a real number and the root is even, the expression underneath the radical is positive. Otherwise, the radical would be imaginary. With this in mind, let's consider the possible values of the variables, $c$ and $d .$

In the radical, the index is even and the exponent of $c$ is even. Therefore, the expression will be real whether the value of $c$ is positive, negative, or equal to $0 .$
In the radical, the index is even and the exponent of $d$ is even. Therefore, the expression will be real whether the value of $d$ is positive, negative, or equal to $0 .$

This means that if we remove

c

or

d

from the radical, we will need absolute value symbols. We can now simplify the given radical by writing the expression inside as powers with exponents equal to the index of the radical.

81 c^{48} d^{64}

Write as a power

9^{2} c^{48} d^{64}

SplitIntoFactors

Split into factors

9^{2} c^{24 \cdot 2} d^{32 \cdot 2}

$a^{m \cdot n} = (a^{m})^{n}$

9^{2} (c^{24})^{2} (d^{32})^{2}

$a^{m} b^{m} = (a b)^{m}$

(9 c^{24} d^{32})^{2}

Calculate root

9 ∣ c^{24} ∣ ∣ d^{32} ∣

$∣ ∣ ∣ c^{24} ∣ ∣ ∣ = c^{24}$

9 c^{24} ∣ d^{32} ∣

$∣ ∣ ∣ d^{32} ∣ ∣ ∣ = d^{32}$

9 c^{24} d^{32}

The simplest form of the expression is

9 c^{24} d^{32} .

Subchapter links

Lesson Check p.370

Practice and Problem-Solving Exercises p.371-372

Standardized Test Prep p.373

Mixed Review p.373