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Chapter 9: Problem 58

Simplify. $$ \sqrt{147} $$

Short Answer

Expert verified
Simplified form: 7\sqrt{3}

Step by step solution

01

Factorize the Number Inside the Square Root

First, find the prime factors of 147. By performing the factorization: 147 is divisible by 3 (since the sum of its digits, 1 + 4 + 7 = 12, is divisible by 3).147 梅 3 = 49.Next, 49 is equal to 7 脳 7.Thus, the prime factorization of 147 is 3 脳 7 脳 7.
02

Group the Factors in Pairs

Group the same factors in pairs to simplify the square root. From the prime factorization 3 脳 7 脳 7, it can be observed that there is one pair of 7's.
03

Simplify the Square Root By Taking Out the Pairs

Since the square root of a pair (7 脳 7) is 7, we can take 7 out of the square root. This leaves us with: \( \sqrt{3 脳 7 脳 7} = 7\sqrt{3} \)

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Key Concepts

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

Prime Factorization
Prime factorization is the process of breaking down a number into smaller, prime components. To determine the prime factors of a number, start by dividing it by the smallest prime number, which is 2. If the number is not divisible by 2, move to the next smallest prime number, 3, and so on. For instance, let's look at 147. Since 147 is not even, it's not divisible by 2, so we check with 3. The sum of the digits of 147 (1 + 4 + 7) is 12, which is divisible by 3. Thus, 147 is also divisible by 3, giving us: 147 梅 3 = 49. Now, 49 is 7 脳 7, hence the prime factorization of 147 is 3, 7, and 7. Prime factorization helps to simplify square roots by revealing paired factors.
Simplifying Radicals
Simplifying radicals involves expressing a square root in its simplest form. Once you have the prime factors of a number, you can identify pairs of factors under the square root. For example, for the number 147, from prime factorization, we have 3 脳 7 脳 7. Simplifying radicals uses these prime factors to rewrite the expression under the root: \ \text{\( \sqrt{147} = \sqrt{3 \times 7 \times 7} \)}. Now, you then look to simplify this expression further by using the properties of square roots.
Pairing Factors
Pairing factors is crucial when simplifying radicals. When you find pairs of the same number in the prime factors inside a square root, you can simplify it by pulling out one of those numbers. For example, in \ \text{\( \sqrt{147} = \sqrt{3 \times 7 \times 7} \)}, we see a pair of 7s. Since the square root of a pair of numbers (e.g., 7 脳 7) is simply the number itself, you can take a 7 out of the square root. This process gives us: \ \text{\( \sqrt{147} = 7\sqrt{3} \)}. Pairing factors simplifies the expression, making it easier to handle and understand.

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Most popular questions from this chapter

(a) Approximate \(\frac{1}{\sqrt{2}}\) by dividing \(\frac{1}{1.414}\) using long division without a calculator. (b) Rationalizing the denominator of \(\frac{1}{\sqrt{2}}\) gives \(\frac{\sqrt{2}}{2}\). Approximate \(\frac{\sqrt{2}}{2}\) by dividing \(\frac{1.414}{2}\) using long division without a calculator. (C) Do you agree that rationalizing the denominator makes calculations easier? Why or why not?

In the following exercises, solve. (a) \(\sqrt{s-8}-s=-8\) (b) \(\sqrt{w+25}-w+5=0\)

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