Question

simplify. express your answer using positive exponents. \\(\frac{(9v^{2}w)(3v^{3}w^{6})}{3vw^{7}}\\)

Explanation:

Step1: Multiply the numerator terms

First, multiply the coefficients and the like - bases in the numerator. For the coefficients: \(9\times3 = 27\). For the \(v\) terms: \(v^{2}\times v^{3}=v^{2 + 3}=v^{5}\) (using the rule \(a^{m}\times a^{n}=a^{m + n}\)). For the \(w\) terms: \(w\times w^{6}=w^{1+6}=w^{7}\) (using the same exponent rule). So the numerator becomes \(27v^{5}w^{7}\).
The expression is now \(\frac{27v^{5}w^{7}}{3vw^{7}}\)

Step2: Divide the coefficients and the like - bases

Divide the coefficients: \(\frac{27}{3}=9\). For the \(v\) terms: \(\frac{v^{5}}{v}=v^{5 - 1}=v^{4}\) (using the rule \(\frac{a^{m}}{a^{n}}=a^{m - n}\)). For the \(w\) terms: \(\frac{w^{7}}{w^{7}}=w^{7 - 7}=w^{0}=1\) (since any non - zero number to the power of 0 is 1).

Answer:

\(9v^{4}\)