Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{-3(q^{-2}a^{5})^{3}}{15q^{8}a^{5}}\\)

Explanation:

Step1: Simplify the numerator's power

Using the power of a product rule \((ab)^n = a^n b^n\) and power of a power rule \((a^m)^n = a^{mn}\), we have:
\((q^{-2}a^{5})^{3}=q^{-2\times3}a^{5\times3}=q^{-6}a^{15}\)
So the numerator becomes \(-3q^{-6}a^{15}\).

Step2: Simplify the coefficient and like bases

First, simplify the coefficient: \(\frac{-3}{15}=-\frac{1}{5}\).
Then, for the \(q\) terms, use the quotient rule \(a^m\div a^n = a^{m - n}\): \(q^{-6}\div q^{8}=q^{-6 - 8}=q^{-14}\).
For the \(a\) terms: \(a^{15}\div a^{5}=a^{15 - 5}=a^{10}\).

Step3: Convert negative exponents to positive

Recall that \(a^{-n}=\frac{1}{a^{n}}\), so \(q^{-14}=\frac{1}{q^{14}}\).
Putting it all together: \(-\frac{1}{5}\times\frac{1}{q^{14}}\times a^{10}=-\frac{a^{10}}{5q^{14}}\).

Answer:

\(-\frac{a^{10}}{5q^{14}}\)