Problem 27 Simplify each radical. $$ \s... [FREE SOLUTION]

91影视

Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 8: Problem 27

Simplify each radical. $$ \sqrt{160} $$

Short Answer

Expert verified

4 $\root{(10)}$

Step by step solution

- Prime Factorization

Find the prime factors of the number under the radical. For 160, the prime factors are: 160 = 2^5 * 5.

- Group the Factors

Group the prime factors into pairs, as each pair can be moved outside the radical. For 160: (2^4) * (2) * (5) = (2^2)^2 * (10).

- Simplify

Move each pair of prime factors outside the radical. So, $\root{160} = \root{(2^4 * 2 * 5)} $. This can be simplified to: 2^2 * $\root{(2 * 5)}$ = 4 $\root{(10)}$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization

Understanding prime factorization is essential to simplifying radicals. It's the process of breaking down a number into its most basic prime number factors. These are numbers that are only divisible by 1 and themselves.

For example, let's break down 160 into prime factors:

Start by dividing by the smallest prime number, which is 2: 160 梅 2 = 80
Then, repeat the process: 80 梅 2 = 40
Continue dividing by 2: 40 梅 2 = 20, 20 梅 2 = 10, and 10 梅 2 = 5
Lastly, 5 is a prime number, so we stop.

So, the prime factorization of 160 is 2^5 * 5. Breaking a number into prime factors helps in grouping these factors, an important step for simplifying radicals.

Grouping Factors

Grouping factors involves pairing up the prime factors so that they can be moved outside the radical. Consider the prime factorization of 160 we determined earlier: 2^5 * 5.

Let's group them:

Write the factors in the order they appear: $(2 * 2 * 2 * 2 * 2 * 5)$
Group them into pairs: $(2^2 * 2^2 * 2 * 5)$
Notice that $2^2$ is a perfect square.

Grouping factors enables us to simplify the expression by moving any complete pairs outside the radical.

Simplifying Square Roots

Simplifying square roots is the final step. After grouping, we can move the paired factors outside the radical.

For 160, we have $(2^2)^2 * 2 * 5$. Because $ (2^2)$ is a perfect square, it comes out as a single 2:

\root{}((2^2)^2 * 2 * 5) = 4 \sqrt{10}

Moving pairs outside the radical helps simplify the expression.
This process can be applied to any square root to make calculations easier and the expression simpler.

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91影视

Simplify each radical. $$ \sqrt{160} $$

Short Answer

Step by step solution

- Prime Factorization

- Group the Factors

- Simplify

Key Concepts

Prime Factorization

Grouping Factors

Simplifying Square Roots

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