Question

find the greatest common factor of these two expressions.\\(27u^{4}v^{2}y^{7}\\) and \\(18u^{5}v^{8}\\)

Explanation:

Step1: Find GCF of coefficients

Prime factorize 27 and 18:
\(27 = 3 \times 3 \times 3\), \(18 = 2 \times 3 \times 3\).
GCF of 27 and 18 is \(3 \times 3 = 9\).

Step2: Find GCF of \(u\)-terms

For \(u^4\) and \(u^5\), the GCF is \(u^{\min(4,5)} = u^4\).

Step3: Find GCF of \(v\)-terms

For \(v^2\) and \(v^8\), the GCF is \(v^{\min(2,8)} = v^2\).

Step4: Find GCF of \(y\)-term (only in first expression)

The second expression has no \(y\)-term, so GCF for \(y\) is \(y^0 = 1\) (not included in final GCF as it doesn't affect).

Step5: Combine all GCFs

Multiply the GCFs of coefficients, \(u\)-terms, and \(v\)-terms:
\(9 \times u^4 \times v^2 = 9u^4v^2\).

Answer:

\(9u^4v^2\)