Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{4s^{8}}{(4s^{-1})^{4}}\\)

Explanation:

Step1: Expand the denominator using power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So for \((4s^{-1})^4\), we have \(4^4\times(s^{-1})^4\).
\(4^4 = 256\) and \((s^{-1})^4 = s^{-4}\) (using the power of a power rule \((a^m)^n=a^{mn}\)), so the denominator becomes \(256s^{-4}\).
The expression is now \(\frac{4s^{8}}{256s^{-4}}\).

Step2: Simplify the coefficient and the variable separately

First, simplify the coefficient: \(\frac{4}{256}=\frac{1}{64}\) (dividing numerator and denominator by 4).
Then, simplify the variable part using the quotient rule for exponents \(a^m\div a^n=a^{m - n}\). So \(s^{8}\div s^{-4}=s^{8-(-4)} = s^{12}\) (because subtracting a negative is adding the positive).

Step3: Combine the simplified coefficient and variable

Multiply the simplified coefficient and the simplified variable part: \(\frac{1}{64}\times s^{12}=\frac{s^{12}}{64}\)

Answer:

\(\frac{s^{12}}{64}\)