Simplify.
$\sqrt{147}$
A. $21 \sqrt{7}$
B. $49 \sqrt{3}$
C. $7 \sqrt{3}$
D. $3 \sqrt{7}$
Answer (1)
Find the prime factorization of 147, which is 3 × 7 2 .
Rewrite the square root as 147 = 3 × 7 2 .
Simplify the square root by taking out the perfect square: 3 × 7 2 = 7 3 .
The simplified form of 147 is 7 3 , so the answer is 7 3 .
Explanation
Understanding the Problem We are asked to simplify the expression 147 . This means we want to find the simplest radical form of the square root of 147.
Finding the Prime Factorization To simplify 147 , we need to find the prime factorization of 147. We can start by dividing 147 by the smallest prime number, 2. Since 147 is odd, it is not divisible by 2. The next prime number is 3. We can divide 147 by 3: 147 = 3 × 49
Complete Prime Factorization Now we need to factor 49. We know that 49 = 7 × 7 = 7 2 . So, the prime factorization of 147 is: 147 = 3 × 7 2
Rewriting the Square Root Now we can rewrite the square root using the prime factorization: 147 = 3 × 7 2
Simplifying the Square Root To simplify the square root, we take out the perfect square ( 7 2 ) from under the radical: 3 × 7 2 = 7 2 × 3 = 7 3
Final Answer So, the simplified form of 147 is 7 3 . Looking at the given options, we see that option C is the correct answer.
Examples
Square roots appear in many contexts, such as calculating distances using the Pythagorean theorem. For example, if you have a right triangle with legs of length 49 and 3 , then the length of the hypotenuse is ( 49 ) 2 + ( 3 ) 2 = 49 + 3 = 52 . Simplifying square roots helps in finding exact values and making calculations easier.
