Question
find the greatest common factor.
6t^4, 9t^4, 9t^3, 15t^4
write your answer as a constant times a product of single variables raised to exponents.
6t^4, 9t^4, 9t^3, 15t^4
write your answer as a constant times a product of single variables raised to exponents.
Answers
GPT-4o mini
To find the greatest common factor (GCF) of the terms \(6t^4\), \(9t^4\), \(9t^3\), and \(15t^4\), we will consider the coefficients and the variable parts separately.
### Step 1: Find the GCF of the coefficients
The coefficients are \(6\), \(9\), \(9\), and \(15\).
- The prime factorization of each number is:
- \(6 = 2 \times 3\)
- \(9 = 3^2\)
- \(15 = 3 \times 5\)
Now, let's find the GCF by taking the lowest power of each prime factor that appears in all the numbers:
- For \(2\): The lowest power is \(0\) (it doesn’t appear in all).
- For \(3\): The lowest power is \(3^1\) (the smallest power of 3 among \(6\), \(9\), and \(15\)).
- For \(5\): The lowest power is \(0\) (it doesn’t appear in \(6\) and \(9\)).
Thus, the GCF of the coefficients is:
\[
GCF(6, 9, 9, 15) = 3^1 = 3.
\]
### Step 2: Find the GCF of the variable parts
The variable parts are \(t^4\), \(t^4\), \(t^3\), and \(t^4\).
The GCF is determined by taking the lowest power of \(t\) present in all terms:
- For \(t\): The lowest power is \(t^3\).
### Step 3: Combine the results
Now, combine the GCF of the coefficients and the GCF of the variable parts:
\[
GCF(6t^4, 9t^4, 9t^3, 15t^4) = 3 \cdot t^3.
\]
### Final Answer
Thus, the greatest common factor is:
\[
3t^3.
\]
### Step 1: Find the GCF of the coefficients
The coefficients are \(6\), \(9\), \(9\), and \(15\).
- The prime factorization of each number is:
- \(6 = 2 \times 3\)
- \(9 = 3^2\)
- \(15 = 3 \times 5\)
Now, let's find the GCF by taking the lowest power of each prime factor that appears in all the numbers:
- For \(2\): The lowest power is \(0\) (it doesn’t appear in all).
- For \(3\): The lowest power is \(3^1\) (the smallest power of 3 among \(6\), \(9\), and \(15\)).
- For \(5\): The lowest power is \(0\) (it doesn’t appear in \(6\) and \(9\)).
Thus, the GCF of the coefficients is:
\[
GCF(6, 9, 9, 15) = 3^1 = 3.
\]
### Step 2: Find the GCF of the variable parts
The variable parts are \(t^4\), \(t^4\), \(t^3\), and \(t^4\).
The GCF is determined by taking the lowest power of \(t\) present in all terms:
- For \(t\): The lowest power is \(t^3\).
### Step 3: Combine the results
Now, combine the GCF of the coefficients and the GCF of the variable parts:
\[
GCF(6t^4, 9t^4, 9t^3, 15t^4) = 3 \cdot t^3.
\]
### Final Answer
Thus, the greatest common factor is:
\[
3t^3.
\]