Find the positive square roots of 5 to the power of 4 x 7 to the power of 2 in prime factorization form

Let's find the positive square roots of \(5^4 \times 7^2\) using prime factorization.

### Step 1: Prime Factorization
The given expression is already in its prime factorized form:

\[
5^4 \times 7^2
\]

### Step 2: Find the Square Root
To find the square root, we take the square root of each prime factor:

\[
\sqrt{5^4 \times 7^2} = \sqrt{5^4} \times \sqrt{7^2}
\]

Now, simplify each square root:

- For \( \sqrt{5^4} \), since \(4\) is even, it simplifies to:

  \[
  \sqrt{5^4} = 5^{4/2} = 5^2
  \]

- For \( \sqrt{7^2} \), since \(2\) is even, it simplifies to:

  \[
  \sqrt{7^2} = 7^{2/2} = 7^1 = 7
  \]

### Step 3: Combine the Simplified Terms
Now, combine the simplified factors:

\[
5^2 \times 7 = 25 \times 7 = 175
\]

Thus, the positive square root of \(5^4 \times 7^2\) is:

\[
\boxed{175}
\]

Let me know if you have any questions or would like more details!

### Related Questions:
1. What are the square roots of \(2^6 \times 3^4\) in prime factorization form?
2. How do you simplify \(\sqrt{4^3 \times 9^2}\) using prime factorization?
3. What is the prime factorization of the square root of \(12 \times 16\)?
4. How do you find the square root of a product of prime numbers raised to powers?
5. What is the square root of \(10^4 \times 2^6\)?

### Tip:
When finding square roots of numbers in prime factorization form, divide the exponent of each prime factor by 2.

Learn how to find the positive square roots of 5^4 x 7^2 using prime factorization. This step-by-step guide covers the process of simplifying square roots and using exponent rules. Suitable for students in Grades 8-10.

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