Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{-2u^{8}}{(3u^{5})^{4}}\\)

Explanation:

Step1: Simplify the denominator using power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So for \((3u^5)^4\), we have \(3^4\times(u^5)^4\). Calculating \(3^4 = 81\) and using the power of a power rule \((a^m)^n=a^{mn}\) for \((u^5)^4\), we get \(u^{5\times4}=u^{20}\). So the denominator becomes \(81u^{20}\).
The expression now is \(\frac{-2u^{8}}{81u^{20}}\)

Step2: Simplify the variable with exponents using quotient rule

The quotient rule for exponents is \(\frac{a^m}{a^n}=a^{m - n}\) (when \(a
eq0\)). Here, for the variable \(u\), we have \(u^{8-20}=u^{- 12}\). And the coefficients remain as \(\frac{-2}{81}\). So the expression is \(\frac{-2}{81}u^{-12}\)

Step3: Convert negative exponent to positive

Recall that \(a^{-n}=\frac{1}{a^n}\) (when \(a
eq0\)). So \(u^{-12}=\frac{1}{u^{12}}\). Substituting this back, we get \(\frac{-2}{81u^{12}}\)

Answer:

\(\frac{-2}{81u^{12}}\)