Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{-6z^{6}}{(-2z)^{3}}\\)

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 \((-2z)^3\), we have \((-2)^3 z^3=-8z^3\). Now the fraction becomes \(\frac{-6z^6}{-8z^3}\).

Step2: Simplify the coefficients and use the quotient rule for exponents

The quotient rule for exponents is \(\frac{a^m}{a^n}=a^{m - n}\) (for \(a
eq0\)). First, simplify the coefficients: \(\frac{-6}{-8}=\frac{3}{4}\). Then, for the variable part \(\frac{z^6}{z^3}\), using the quotient rule, we get \(z^{6-3}=z^3\).

Step3: Combine the results

Multiply the simplified coefficient and the simplified variable part: \(\frac{3}{4}\times z^3=\frac{3z^3}{4}\).

Answer:

\(\frac{3z^3}{4}\)