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Answer :
To find the prime factorization of [tex]\(175 x^2 y^3\)[/tex], we can break it down into simple steps. Let's do this step by step:
1. Prime Factorization of 175:
- First, consider 175. We need to find its prime factors.
- We start with the smallest prime number, 2. Since 175 is odd, it's not divisible by 2.
- Next, try dividing by 3. The sum of the digits of 175 is 13 (1 + 7 + 5), which is not divisible by 3, so 175 is not divisible by 3.
- Try 5 next. Since 175 ends in a 5, it is divisible by 5.
- Dividing 175 by 5 gives 35. So, 175 = 5 × 35.
- Next, find the factors of 35. Again, since 35 ends in a 5, it is divisible by 5.
- Dividing 35 by 5 gives 7. So, 35 = 5 × 7.
- Combining these factors, we have 175 = 5 × 5 × 7.
2. Combining with the Variables:
- Now consider the variable parts: [tex]\(x^2\)[/tex] and [tex]\(y^3\)[/tex].
- The expression [tex]\(x^2\)[/tex] means we have x multiplied by x, or [tex]\(x \cdot x\)[/tex].
- The expression [tex]\(y^3\)[/tex] means we have y multiplied by y multiplied by y, or [tex]\(y \cdot y \cdot y\)[/tex].
3. Final Prime Factorization:
- Combine the prime factors of 175 with the variables.
- This gives us: [tex]\(5 \cdot 5 \cdot 7 \cdot x \cdot x \cdot y \cdot y \cdot y\)[/tex].
So, the prime factorization of [tex]\(175 x^2 y^3\)[/tex] is:
[tex]\[ 5 \cdot 5 \cdot 7 \cdot x \cdot x \cdot y \cdot y \cdot y \][/tex]
This matches the correct choice in the given options.
1. Prime Factorization of 175:
- First, consider 175. We need to find its prime factors.
- We start with the smallest prime number, 2. Since 175 is odd, it's not divisible by 2.
- Next, try dividing by 3. The sum of the digits of 175 is 13 (1 + 7 + 5), which is not divisible by 3, so 175 is not divisible by 3.
- Try 5 next. Since 175 ends in a 5, it is divisible by 5.
- Dividing 175 by 5 gives 35. So, 175 = 5 × 35.
- Next, find the factors of 35. Again, since 35 ends in a 5, it is divisible by 5.
- Dividing 35 by 5 gives 7. So, 35 = 5 × 7.
- Combining these factors, we have 175 = 5 × 5 × 7.
2. Combining with the Variables:
- Now consider the variable parts: [tex]\(x^2\)[/tex] and [tex]\(y^3\)[/tex].
- The expression [tex]\(x^2\)[/tex] means we have x multiplied by x, or [tex]\(x \cdot x\)[/tex].
- The expression [tex]\(y^3\)[/tex] means we have y multiplied by y multiplied by y, or [tex]\(y \cdot y \cdot y\)[/tex].
3. Final Prime Factorization:
- Combine the prime factors of 175 with the variables.
- This gives us: [tex]\(5 \cdot 5 \cdot 7 \cdot x \cdot x \cdot y \cdot y \cdot y\)[/tex].
So, the prime factorization of [tex]\(175 x^2 y^3\)[/tex] is:
[tex]\[ 5 \cdot 5 \cdot 7 \cdot x \cdot x \cdot y \cdot y \cdot y \][/tex]
This matches the correct choice in the given options.
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