Question

simplify: \\(\sqrt{\frac{25}{441}}\\)\
the prime factorization of 25 is \
the prime factorization of 441 is \
the expression \\(\sqrt{\frac{25}{441}}\\) in simple \
2(5)\
5(5)\
1(25)

Explanation:

Part 1: Prime Factorization of 25

Step 1: Recall Prime Factorization

Prime factorization involves expressing a number as a product of prime numbers. For 25, we know that \( 5\times5 = 25 \), and 5 is a prime number. So the prime factorization of 25 is \( 5\times5 \) or \( 5(5) \).

Step 1: Start with Small Primes

We start by dividing 441 by the smallest prime numbers. First, check divisibility by 3: \( 4 + 4 + 1=9 \), which is divisible by 3. So \( 441\div3 = 147 \).

Step 2: Divide 147 by 3

\( 147\div3 = 49 \).

Step 3: Factorize 49

We know that \( 49 = 7\times7 \). So putting it all together, \( 441=3\times3\times7\times7 \) or \( 3(3)\times7(7) \) (or in boxed form, but as per instruction, we can write it as \( 3\times3\times7\times7 \) or \( 3^2\times7^2 \)). But if we follow the format similar to the first part, we can write it as \( 3(3)\times7(7) \) or more concisely, since the options might be in terms of prime factors multiplied, but since the first part was in the form of a product of the same prime (for 25, 5 and 5), for 441, the prime factorization is \( 3\times3\times7\times7 \), which can be written as \( 3(3)\times7(7) \), but if we consider the structure, maybe the intended answer is \( 3\times3\times7\times7 \) or \( 3^2\times7^2 \). However, if we follow the pattern of the first dropdown (which had 2(5), 5(5), 1(25)), for 441, the prime factors are 3, 3, 7, 7, so the correct way to write the prime factorization is \( 3\times3\times7\times7 \) or \( 3(3)\times7(7) \). But maybe the options are in a similar format, so the prime factorization of 441 is \( 3\times3\times7\times7 \) (or \( 3^2\times7^2 \)).

Step 1: Use Property of Square Roots

We know that \( \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}} \) for \( a\geq0 \) and \( b > 0 \). So \( \sqrt{\frac{25}{441}}=\frac{\sqrt{25}}{\sqrt{441}} \).

Step 2: Simplify Square Roots

We know that \( \sqrt{25} = 5 \) (since \( 5\times5 = 25 \)) and \( \sqrt{441}=21 \) (since \( 21\times21 = 441 \)). So \( \frac{\sqrt{25}}{\sqrt{441}}=\frac{5}{21} \).

Answer:

5(5)

Part 2: Prime Factorization of 441