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Let's work through the two parts of the problem using the prime factorization method to find the square roots.

### (a) Find the square root of \( 9 \frac{67}{121} \)

This is a mixed fraction, \( 9 \frac{67}{121} \), which we first convert into an improper fraction:

\[
9 \frac{67}{121} = \frac{9 \times 121 + 67}{121} = \frac{1089 + 67}{121} = \frac{1156}{121}
\]

Now, to find the square root of \( \frac{1156}{121} \), we take the square root of the numerator and the denominator separately:

\[
\sqrt{\frac{1156}{121}} = \frac{\sqrt{1156}}{\sqrt{121}} = \frac{34}{11}
\]

Thus, the square root of \( 9 \frac{67}{121} \) is \( \frac{34}{11} \).

### (b) Find the square root of \( 0.4225 \)

We convert \( 0.4225 \) into a fraction:

\[
0.4225 = \frac{4225}{10000}
\]

Now, using prime factorization:

1. Prime factorization of 4225: \( 4225 = 65^2 = 5^2 \times 13^2 \)
2. Prime factorization of 10000: \( 10000 = (10^4) = (2^4 \times 5^4) \)

Taking the square root of both the numerator and denominator:

\[
\sqrt{\frac{4225}{10000}} = \frac{\sqrt{4225}}{\sqrt{10000}} = \frac{65}{100}
\]

Thus, the square root of \( 0.4225 \) is \( \frac{65}{100} = 0.65 \).

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Would you like further clarification on any steps or more details? Here are some related questions to deepen your understanding:

1. How can prime factorization help in simplifying the square roots of large numbers?
2. Why do we take the square root of both the numerator and denominator separately in fractional square roots?
3. Can all square roots of decimal numbers be expressed as fractions?
4. How would you approach finding the square root of a fraction that doesn’t simplify easily?
5. What are some other methods for finding square roots besides prime factorization?

**Tip**: Always check if you can convert a mixed fraction into an improper fraction before solving for roots—it makes the process clearer!

Using prime factorization method, find the square root of: 9 67/121 and 0.4225.

Learn how to find the square roots of 9 67/121 and 0.4225 using the prime factorization method. This step-by-step solution covers key mathematical concepts like square roots, prime factorization, and fractional square roots. Suitable for students in Grades 8-10.

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