Question

simplify. express your answer using positive exponents.\\(\frac{2u^{6}vw^{4}}{2u^{7}vwcdot u^{6}vw^{2}}\\)

Explanation:

Step1: Simplify the denominator by multiplying the terms

First, we handle the denominator \(2u^{7}vw\cdot u^{6}vw^{2}\). When multiplying terms with the same base, we add the exponents. For the coefficient, it's \(2\) (since there's only one coefficient of \(2\) and the other term has a coefficient of \(1\) for \(u^{6}vw^{2}\)). For the \(u\) terms: \(u^{7}\cdot u^{6}=u^{7 + 6}=u^{13}\). For the \(v\) terms: \(v\cdot v = v^{1+1}=v^{2}\). For the \(w\) terms: \(w\cdot w^{2}=w^{1 + 2}=w^{3}\). So the denominator simplifies to \(2u^{13}v^{2}w^{3}\).

The original expression is now \(\frac{2u^{6}vw^{4}}{2u^{13}v^{2}w^{3}}\)

Step2: Simplify the coefficients and each variable separately

• Coefficients: \(\frac{2}{2}=1\)
• For \(u\) terms: Using the rule \(\frac{a^{m}}{a^{n}}=a^{m - n}\), we have \(\frac{u^{6}}{u^{13}}=u^{6-13}=u^{-7}\). But we need positive exponents, so \(u^{-7}=\frac{1}{u^{7}}\)
• For \(v\) terms: \(\frac{v}{v^{2}}=v^{1 - 2}=v^{-1}=\frac{1}{v}\)
• For \(w\) terms: \(\frac{w^{4}}{w^{3}}=w^{4 - 3}=w^{1}=w\)

Step3: Combine the simplified terms

Multiply all the simplified parts together: \(1\times\frac{1}{u^{7}}\times\frac{1}{v}\times w=\frac{w}{u^{7}v}\)

Answer:

\(\frac{w}{u^{7}v}\)