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Chapter 5: Problem 25

Simplify. $$ \sqrt{147} $$

Short Answer

Expert verified
7鈭�3

Step by step solution

01

- Prime Factorization

Find the prime factors of 147. First, divide 147 by the smallest prime number 3: 147 梅 3 = 49. Next, factor 49 into primes: 49 = 7 脳 7. Therefore, the prime factorization of 147 is 3 脳 7 脳 7.
02

- Group the Factors

Rewrite the radical expression using the prime factors: \[ \sqrt{147} = \sqrt{3 \times 7 \times 7} \].
03

- Simplify the Radical

Group the factors in pairs under the square root: \[ \sqrt{3 \times 7 \times 7} = \sqrt{7 \times 7} \times \sqrt{3} \]. The square root of a squared term pairs can be simplified as: \[ \sqrt{7 \times 7} = 7 \]. Now, multiply the simplified part outside the root with what remains inside: \[ 7 \sqrt{3} \].

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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 composite number into its prime number factors. Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves.
To factorize 147, we start by dividing by the smallest prime number possible. Here, 147 梅 3 = 49.
Next, we factor 49 into primes, which gives us 49 = 7 脳 7.
Therefore, the prime factors of 147 are 3, 7, and 7. We can write this as: 147 = 3 脳 7 脳 7.
Understanding this makes it easier to simplify radical expressions.
Radical Expressions
Radical expressions involve roots, like square roots, and represent a value that, when multiplied by itself, gives the original number.
For instance, \( \sqrt{147} \) is a radical expression.
Knowing the prime factors, we can rewrite the expression as: \( \sqrt{147} = \sqrt{3 \times 7 \times 7} \).
This is crucial as it allows us to simplify the radical.
The goal in simplifying radicals is to identify pairs or groups of numbers whose square roots we know.
Square Roots
The square root of a number is a value that, when squared, returns the original number. Square roots are often written with the radical symbol: \( \sqrt{} \).
For example, \( \sqrt{49} = 7 \) because 7 脳 7 = 49.
When simplifying \( \sqrt{147} \), we use its prime factors: \( \sqrt{3 \times 7 \times 7} \).
Grouping the factors inside the radical, we get: \( \sqrt{7 \times 7} \times \sqrt{3} = 7 \sqrt{3} \).
Here, \( \sqrt{7 \times 7} = 7 \) can be taken out of the radical.
Thus, \( \sqrt{147} \) simplifies to \( 7 \sqrt{3}\).

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