Question

(\frac{f^{0}g^{5}cdot f^{-3}g}{f^{-1}g^{-5}})

Explanation:

Step1: Simplify numerator (exponent rule for multiplication)

For the numerator \( f^{0.5} \cdot f^{-3}g \), use \( a^m \cdot a^n = a^{m + n} \) for \( f \)-terms:
\( f^{0.5 + (-3)}g = f^{-2.5}g \)

Step2: Simplify denominator (exponent rule for division)

For the denominator \( f^{-1}g^{-5} \), use \( \frac{a^m}{a^n} = a^{m - n} \) for \( f \)-terms and \( g \)-terms:
\( \frac{f^{-2.5}g}{f^{-1}g^{-5}} = f^{-2.5 - (-1)} \cdot g^{1 - (-5)} \)

Step3: Calculate exponents

For \( f \)-exponent: \( -2.5 + 1 = -1.5 \) (or \( -\frac{3}{2} \))
For \( g \)-exponent: \( 1 + 5 = 6 \)
So the expression becomes \( f^{-1.5}g^{6} \) or \( \frac{g^{6}}{f^{1.5}} \) (since \( a^{-n} = \frac{1}{a^n} \))

Answer:

\( \boldsymbol{\frac{g^{6}}{f^{\frac{3}{2}}}} \) (or \( \boldsymbol{f^{-1.5}g^{6}} \))