Simplify. [tex]\sqrt{147}[/tex]
A. [tex]21 \sqrt{7}[/tex]
B. [tex]49 \sqrt{3}[/tex]
C. [tex]7 \sqrt{3}[/tex]
D. [tex]3 \sqrt{7}[/tex]
Answer (2)
147 can be simplified by finding the prime factorization of 147. The prime factorization of 147 is 3 × 7 2 . Rewrite the square root as 147 = 3 × 7 2 . Simplify the square root to get 7 3 . The final answer is 7 3 .
Explanation
Understanding the Problem We are asked to simplify 147 . To do this, we need to find the prime factorization of 147 and look for perfect square factors.
Prime Factorization First, we find the prime factorization of 147. We can see that 147 is divisible by 3 since the sum of its digits (1+4+7=12) is divisible by 3. Dividing 147 by 3, we get 49. So, 147 = 3 × 49 .
Rewriting the Square Root Now, we know that 49 is a perfect square, 49 = 7 2 . Therefore, the prime factorization of 147 is 147 = 3 × 7 2 .
Simplifying the Square Root We can rewrite the square root as follows: 147 = 3 × 7 2 = 7 2 × 3 = 7 3 So, the simplified form of 147 is 7 3 .
Final Answer Comparing our simplified expression with the given options, we see that option C, 7 3 , 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 48 and 3 , the length of the hypotenuse is ( 48 ) 2 + ( 3 ) 2 = 48 + 3 = 51 . Simplifying square roots helps in expressing these lengths in simplest form, making further calculations easier.
The simplified form of 147 is 7 3 . The process involves finding the prime factorization of 147, which leads to separating perfect squares and simplifying the square root. Therefore, the correct choice is option C: 7 3 .
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