Question

simplify the expression. assume that the denominator does not equal zero. write any variables in alphabetical order. \\(\frac{-12c^{3}q^{0}f^{-2}}{6c^{5}q^{-3}f^{4}}\\)

Explanation:

Step1: Simplify the coefficient and the constant term of \( q \)

First, simplify the coefficient \(\frac{-12}{6} = -2\). And \( q^0 = 1 \) (any non - zero number to the power of 0 is 1). So the expression becomes \(-2\times\frac{c^{3}}{c^{5}}\times\frac{1}{d^{-3}}\times\frac{f^{-2}}{f^{4}}\)

Step2: Use the quotient rule of exponents \( \frac{a^{m}}{a^{n}}=a^{m - n} \) for \( c \), \( d \) and \( f \)

For the variable \( c \): \(\frac{c^{3}}{c^{5}}=c^{3-5}=c^{-2}=\frac{1}{c^{2}}\) (by the rule \( a^{-n}=\frac{1}{a^{n}} \))
For the variable \( d \): \(\frac{1}{d^{-3}}=d^{3}\) (by the rule \( \frac{1}{a^{-n}}=a^{n} \))
For the variable \( f \): \(\frac{f^{-2}}{f^{4}}=f^{-2 - 4}=f^{-6}=\frac{1}{f^{6}}\)

Step3: Combine all the simplified parts

Multiply all the parts together: \(-2\times\frac{1}{c^{2}}\times d^{3}\times\frac{1}{f^{6}}=\frac{-2d^{3}}{c^{2}f^{6}}\)

Answer:

\(\frac{-2d^{3}}{c^{2}f^{6}}\)