Question

simplify.
\\(\frac{4u v^{7}}{u^{5} v^{3}}\\)

Explanation:

Step1: Simplify the coefficient and the \(u\)-terms

For the coefficient, we have \(4\) (which remains as is for now). For the \(u\)-terms, we use the rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\). Here, for \(u\), we have \(\frac{u}{u^5}=u^{1-5}=u^{-4}=\frac{1}{u^4}\) (but we can also handle it as part of the overall simplification by subtracting exponents directly in the fraction).
For the \(v\)-terms, using the same exponent rule \(\frac{v^7}{v^3}=v^{7 - 3}=v^4\).
Putting it all together, we have:
\[
\frac{4uv^7}{u^5v^3}=4\times\frac{u}{u^5}\times\frac{v^7}{v^3}
\]

Step2: Apply exponent rules

Simplify the \(u\) and \(v\) terms:
\[
4\times u^{1 - 5}\times v^{7 - 3}=4\times u^{-4}\times v^{4}
\]
But we can also write this with positive exponents for \(u\) by moving it to the denominator:
\[
\frac{4v^4}{u^4}
\]
(Alternatively, we can think of it as subtracting the exponents of like bases directly: for \(u\), exponent of numerator is \(1\), denominator is \(5\), so \(1-5=-4\); for \(v\), \(7 - 3 = 4\); coefficient is \(4\). So combining, we get \(4u^{-4}v^{4}=\frac{4v^4}{u^4}\))

Answer:

\(\frac{4v^4}{u^4}\)