Question

perform the indicated operations. write the answer using only positive exponents. assume all variables represent nonzero real numbers.\\(\frac{(3p^2q)q^3}{15p^6q^7}\\)\\(\frac{(3p^2q)q^3}{15p^6q^7}=\square\\)(simplify your answer. use integers or fractions for any numbers in the expression)

Explanation:

Step1: Simplify the numerator

First, we simplify the numerator \((3p^{2}q)q^{3}\). Using the rule of exponents \(a^{m}\cdot a^{n}=a^{m + n}\) for the variable \(q\), we have:
\((3p^{2}q)q^{3}=3p^{2}(q\cdot q^{3}) = 3p^{2}q^{1 + 3}=3p^{2}q^{4}\)

Step2: Simplify the fraction

Now we have the fraction \(\frac{3p^{2}q^{4}}{15p^{6}q^{7}}\). We can simplify the coefficients and the variables separately.

• For the coefficients: \(\frac{3}{15}=\frac{1}{5}\)
• For the variable \(p\): Using the rule \(\frac{a^{m}}{a^{n}}=a^{m - n}\), we have \(\frac{p^{2}}{p^{6}}=p^{2-6}=p^{- 4}=\frac{1}{p^{4}}\) (since we want positive exponents)
• For the variable \(q\): Using the same rule, \(\frac{q^{4}}{q^{7}}=q^{4 - 7}=q^{-3}=\frac{1}{q^{3}}\)

Now we multiply these simplified parts together:
\(\frac{1}{5}\times\frac{1}{p^{4}}\times\frac{1}{q^{3}}=\frac{1}{5p^{4}q^{3}}\)

Answer:

\(\frac{1}{5p^{4}q^{3}}\)