Question

Find the greatest common factor. 6t4, 9t4, 9t3, 15t4 Write your answer as a constant times a product of single variables raised to exponents.

Answer 1

To find the greatest common factor (GCF) of the given expressions \(6t^4\), \(9t^4\), \(9t^3\), and \(15t^4\), we will look at both the numerical coefficients and the variable parts.

1. **Numerical Coefficients:**
- The coefficients are \(6\), \(9\), \(9\), and \(15\).
- To find the GCF of these numbers, we can factor them:
- \(6 = 2 \times 3\)
- \(9 = 3^2\)
- \(15 = 3 \times 5\)
- The common factors are determined by taking the minimum power of each prime factor:
- The only prime factor present in all is \(3\).
- The minimum power of \(3\) in the factorizations is \(3^1\).
- Therefore, the GCF of the coefficients is \(3\).

2. **Variable Parts:**
- The variable parts are \(t^4\), \(t^4\), \(t^3\), and \(t^4\).
- The GCF for the variable part is determined by the lowest exponent of \(t\) in all terms:
- The minimum exponent of \(t\) is \(3\).
- Thus, the GCF for the variable part is \(t^3\).

3. **Combining the GCFs:**
- The overall GCF is the product of the GCF of the coefficients and the GCF of the variables.
- Therefore, the GCF is:
\[
3t^3
\]

So, the greatest common factor of \(6t^4\), \(9t^4\), \(9t^3\), and \(15t^4\) is \(\boxed{3t^3}\).