Answer :
To find the greatest common factor (GCF) of the expressions [tex]\(60x^4y^7\)[/tex], [tex]\(45x^5y^5\)[/tex], and [tex]\(75x^3y\)[/tex], follow these steps:
1. Find the GCF of the coefficients:
- Take the coefficients of each term: 60, 45, and 75.
- Determine the GCF of these numbers:
- The prime factorization of 60 is [tex]\(2^2 \times 3 \times 5\)[/tex].
- The prime factorization of 45 is [tex]\(3^2 \times 5\)[/tex].
- The prime factorization of 75 is [tex]\(3 \times 5^2\)[/tex].
- The common factors are [tex]\(3\)[/tex] and [tex]\(5\)[/tex].
- The GCF of the coefficients is [tex]\(3 \times 5 = 15\)[/tex].
2. Determine the GCF of the variables with respect to [tex]\(x\)[/tex]:
- Look at the powers of [tex]\(x\)[/tex] in each expression: [tex]\(x^4\)[/tex], [tex]\(x^5\)[/tex], and [tex]\(x^3\)[/tex].
- Take the smallest power: [tex]\(\min(4, 5, 3) = 3\)[/tex].
- Thus, the GCF for [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].
3. Determine the GCF of the variables with respect to [tex]\(y\)[/tex]:
- Look at the powers of [tex]\(y\)[/tex] in each expression: [tex]\(y^7\)[/tex], [tex]\(y^5\)[/tex], and [tex]\(y^1\)[/tex].
- Take the smallest power: [tex]\(\min(7, 5, 1) = 1\)[/tex].
- Thus, the GCF for [tex]\(y\)[/tex] is [tex]\(y^1\)[/tex] (often written simply as [tex]\(y\)[/tex]).
4. Combine the results:
- Combine the GCF of the coefficients and variables to form the complete GCF of the expressions.
- The GCF is [tex]\(15x^3y\)[/tex].
Therefore, the greatest common factor of [tex]\(60x^4y^7\)[/tex], [tex]\(45x^5y^5\)[/tex], and [tex]\(75x^3y\)[/tex] is [tex]\(15x^3y\)[/tex].
1. Find the GCF of the coefficients:
- Take the coefficients of each term: 60, 45, and 75.
- Determine the GCF of these numbers:
- The prime factorization of 60 is [tex]\(2^2 \times 3 \times 5\)[/tex].
- The prime factorization of 45 is [tex]\(3^2 \times 5\)[/tex].
- The prime factorization of 75 is [tex]\(3 \times 5^2\)[/tex].
- The common factors are [tex]\(3\)[/tex] and [tex]\(5\)[/tex].
- The GCF of the coefficients is [tex]\(3 \times 5 = 15\)[/tex].
2. Determine the GCF of the variables with respect to [tex]\(x\)[/tex]:
- Look at the powers of [tex]\(x\)[/tex] in each expression: [tex]\(x^4\)[/tex], [tex]\(x^5\)[/tex], and [tex]\(x^3\)[/tex].
- Take the smallest power: [tex]\(\min(4, 5, 3) = 3\)[/tex].
- Thus, the GCF for [tex]\(x\)[/tex] is [tex]\(x^3\)[/tex].
3. Determine the GCF of the variables with respect to [tex]\(y\)[/tex]:
- Look at the powers of [tex]\(y\)[/tex] in each expression: [tex]\(y^7\)[/tex], [tex]\(y^5\)[/tex], and [tex]\(y^1\)[/tex].
- Take the smallest power: [tex]\(\min(7, 5, 1) = 1\)[/tex].
- Thus, the GCF for [tex]\(y\)[/tex] is [tex]\(y^1\)[/tex] (often written simply as [tex]\(y\)[/tex]).
4. Combine the results:
- Combine the GCF of the coefficients and variables to form the complete GCF of the expressions.
- The GCF is [tex]\(15x^3y\)[/tex].
Therefore, the greatest common factor of [tex]\(60x^4y^7\)[/tex], [tex]\(45x^5y^5\)[/tex], and [tex]\(75x^3y\)[/tex] is [tex]\(15x^3y\)[/tex].
