Question

perform the indicated operations. write the answer using only positive exponents. assume all variables represent nonzero real numbers.
\\(\frac{(4p^2q)q^3}{6p^6q^6}\\)
\\(\frac{(4p^2q)q^3}{6p^6q^6} = \square\\)
(simplify your answer. use integers or fractions for any numbers in the expression)

Explanation:

Step1: Simplify the numerator using power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\) and \((a^m)^n=a^{mn}\), and \(a^m\times a^n = a^{m + n}\).
First, simplify \((4p^{2}q)^{3}\times q^{3}\):
\((4p^{2}q)^{3}=4^{3}\times(p^{2})^{3}\times q^{3}=64p^{6}q^{3}\)
Then multiply by \(q^{3}\): \(64p^{6}q^{3}\times q^{3}=64p^{6}q^{3 + 3}=64p^{6}q^{6}\)
So the numerator becomes \(64p^{6}q^{6}\)

Step2: Divide the numerator by the denominator

The expression is now \(\frac{64p^{6}q^{6}}{6p^{6}q^{6}}\)
We can use the rule \(\frac{a^m}{a^m}=1\) (for \(a
eq0\)) and \(\frac{a}{b}=\frac{ac}{bc}\) (simplification of fractions).
For the coefficients: \(\frac{64}{6}=\frac{32}{3}\)
For the \(p\) terms: \(\frac{p^{6}}{p^{6}} = 1\)
For the \(q\) terms: \(\frac{q^{6}}{q^{6}}=1\)
Multiplying these results together: \(\frac{32}{3}\times1\times1=\frac{32}{3}\)

Answer:

\(\frac{32}{3}\)