Question
Find the greatest common factor. \[ 2 n^{3}, 14 n^{3}, 2 n^{2} \]
Ask by Schofield Long. in the United States
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The greatest common factor is \(2n^2\).
Solution
1. Write down the given expressions:
\[
2n^3,\quad 14n^3,\quad 2n^2.
\]
2. Find the greatest common factor (GCF) for the numerical coefficients:
- The coefficients are \(2\), \(14\), and \(2\).
- The GCF of \(2\), \(14\), and \(2\) is \(2\).
3. Find the GCF for the variable \(n\):
- The exponents of \(n\) are \(3\), \(3\), and \(2\).
- The minimum exponent is \(2\), so the GCF for \(n\) is \(n^2\).
4. Multiply the GCF of the coefficients and the GCF of the variables:
\[
\text{GCF} = 2 \cdot n^2 = 2n^2.
\]
Thus, the greatest common factor of the expressions is \(2n^2\).
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Simplify this solution Extra Insights
To find the greatest common factor (GCF) of the terms \(2 n^{3}\), \(14 n^{3}\), and \(2 n^{2}\), we first determine the coefficients and the variable parts separately. The coefficients are \(2\), \(14\), and \(2\). The GCF of these numbers is \(2\). Now, we look at the variable part. The lowest power of \(n\) among \(n^{3}\), \(n^{3}\), and \(n^{2}\) is \(n^{2}\). Combining these, the greatest common factor is: \[ \text{GCF} = 2 n^{2} \]
