Question
Find the greatest common factor for the list of monomials.
20 x Superscript 4 Baseline y Superscript 4 Baseline z, 4 x Superscript 4 Baseline y Superscript 4, 60 xy Superscript 4
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Part 1
The greatest common factor is
enter your response here.
20 x Superscript 4 Baseline y Superscript 4 Baseline z, 4 x Superscript 4 Baseline y Superscript 4, 60 xy Superscript 4
Question content area bottom
Part 1
The greatest common factor is
enter your response here.
Answers
GPT-4o mini
To find the greatest common factor (GCF) of the monomials \( 20 x^4 y^4 z \), \( 4 x^4 y^4 \), and \( 60 xy^4 \), we will first find the coefficients and the variables separately.
### Step 1: Find the GCF of the coefficients.
- The coefficients of the monomials are:
- 20
- 4
- 60
To find the GCF of these coefficients:
- The prime factorization of 20 is \( 2^2 \cdot 5 \).
- The prime factorization of 4 is \( 2^2 \).
- The prime factorization of 60 is \( 2^2 \cdot 3 \cdot 5 \).
The GCF of the coefficients is the product of the lowest powers of all prime factors:
- For \( 2 \): the lowest power is \( 2^2 \).
- For \( 3 \): it appears in 60 only, so we take \( 3^0 \).
- For \( 5 \): it appears in 20 and 60, but not in 4, so we take \( 5^0 \).
Thus, the GCF of the coefficients is:
\[
2^2 = 4
\]
### Step 2: Find the GCF of the variable parts.
Now we'll find the GCF for each variable present in the monomials:
- For \( x \):
- The degrees are \( 4 \) (from \( 20 x^4 y^4 z \)), \( 4 \) (from \( 4 x^4 y^4 \)), and \( 1 \) (from \( 60 xy^4 \)).
- The minimum degree is \( 1 \).
- Therefore, the GCF for \( x \) is \( x^1 = x \).
- For \( y \):
- The degrees are \( 4 \) (from \( 20 x^4 y^4 z \)), \( 4 \) (from \( 4 x^4 y^4 \)), and \( 4 \) (from \( 60 xy^4 \)).
- The minimum degree is \( 4 \).
- Therefore, the GCF for \( y \) is \( y^4 \).
- For \( z \):
- The degrees are \( 1 \) (from \( 20 x^4 y^4 z \)), \( 0 \) (from \( 4 x^4 y^4 \)), and \( 0 \) (from \( 60 xy^4 \)).
- The minimum degree is \( 0 \).
- Therefore, the GCF for \( z \) is \( z^0 = 1 \) (we do not include \( z \) in the GCF).
### Step 3: Combine the GCFs.
Now we combine the GCFs for the coefficients and variables:
\[
\text{GCF} = 4 \cdot x^1 \cdot y^4 = 4xy^4
\]
### Conclusion
The greatest common factor of the given monomials is:
\[
\boxed{4xy^4}
\]
### Step 1: Find the GCF of the coefficients.
- The coefficients of the monomials are:
- 20
- 4
- 60
To find the GCF of these coefficients:
- The prime factorization of 20 is \( 2^2 \cdot 5 \).
- The prime factorization of 4 is \( 2^2 \).
- The prime factorization of 60 is \( 2^2 \cdot 3 \cdot 5 \).
The GCF of the coefficients is the product of the lowest powers of all prime factors:
- For \( 2 \): the lowest power is \( 2^2 \).
- For \( 3 \): it appears in 60 only, so we take \( 3^0 \).
- For \( 5 \): it appears in 20 and 60, but not in 4, so we take \( 5^0 \).
Thus, the GCF of the coefficients is:
\[
2^2 = 4
\]
### Step 2: Find the GCF of the variable parts.
Now we'll find the GCF for each variable present in the monomials:
- For \( x \):
- The degrees are \( 4 \) (from \( 20 x^4 y^4 z \)), \( 4 \) (from \( 4 x^4 y^4 \)), and \( 1 \) (from \( 60 xy^4 \)).
- The minimum degree is \( 1 \).
- Therefore, the GCF for \( x \) is \( x^1 = x \).
- For \( y \):
- The degrees are \( 4 \) (from \( 20 x^4 y^4 z \)), \( 4 \) (from \( 4 x^4 y^4 \)), and \( 4 \) (from \( 60 xy^4 \)).
- The minimum degree is \( 4 \).
- Therefore, the GCF for \( y \) is \( y^4 \).
- For \( z \):
- The degrees are \( 1 \) (from \( 20 x^4 y^4 z \)), \( 0 \) (from \( 4 x^4 y^4 \)), and \( 0 \) (from \( 60 xy^4 \)).
- The minimum degree is \( 0 \).
- Therefore, the GCF for \( z \) is \( z^0 = 1 \) (we do not include \( z \) in the GCF).
### Step 3: Combine the GCFs.
Now we combine the GCFs for the coefficients and variables:
\[
\text{GCF} = 4 \cdot x^1 \cdot y^4 = 4xy^4
\]
### Conclusion
The greatest common factor of the given monomials is:
\[
\boxed{4xy^4}
\]