Question

express the following fraction in simplest form, only using positive exponents. \\(\frac{-3v^{9}}{(-4v^{-3})^{4}}\\)

Explanation:

Step1: Simplify the denominator's power

First, we simplify \((-4v^{-3})^4\). Using the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((a^m)^n=a^{mn}\), we have:
\((-4)^4\times(v^{-3})^4 = 256\times v^{-12}=\frac{256}{v^{12}}\)

Step2: Rewrite the fraction

Now the original fraction \(\frac{-3v^{9}}{(-4v^{-3})^4}\) becomes \(\frac{-3v^{9}}{\frac{256}{v^{12}}}\). Dividing by a fraction is the same as multiplying by its reciprocal, so this is \(-3v^{9}\times\frac{v^{12}}{256}\).

Step3: Multiply the variables

Using the rule of exponents \(a^m\times a^n=a^{m + n}\) for the \(v\) terms: \(v^{9}\times v^{12}=v^{9 + 12}=v^{21}\). So now we have \(\frac{-3v^{21}}{256}\).

Answer:

\(\frac{-3v^{21}}{256}\)