Question

simplify the given expression.
\\(\frac{3p^4}{5q^5(r - 5)^3} \cdot \frac{37q^2(r - 5)}{30p^3}\\)
\\(\bigcirc\\) \\(\frac{37p}{50q^2(r - 5)^2}\\)
\\(\bigcirc\\) \\(\frac{37p(r - 5)^2}{50q^3}\\)
\\(\bigcirc\\) \\(\frac{37p}{50q^3(r - 5)^2}\\)
\\(\bigcirc\\) \\(\frac{37p^2}{50q^3(r - 5)^2}\\)

Explanation:

Step1: Multiply numerators and denominators

Multiply the numerators together: \(3p^4 \cdot 37q^2(r - 5)=111p^4q^2(r - 5)\)
Multiply the denominators together: \(5q^5(r - 5)^3 \cdot 30p^3 = 150p^3q^5(r - 5)^3\)
So the expression becomes \(\frac{111p^4q^2(r - 5)}{150p^3q^5(r - 5)^3}\)

Step2: Simplify coefficients

Simplify \(\frac{111}{150}\) by dividing numerator and denominator by 3: \(\frac{111\div3}{150\div3}=\frac{37}{50}\)

Step3: Simplify \(p\) terms

Using the rule \(a^m\div a^n=a^{m - n}\), for \(p\) terms: \(p^4\div p^3 = p^{4 - 3}=p\)

Step4: Simplify \(q\) terms

For \(q\) terms: \(q^2\div q^5 = q^{2 - 5}=q^{- 3}=\frac{1}{q^3}\) (or directly subtract exponents in the denominator: \(q^{5 - 2}=q^3\) in the denominator)

Step5: Simplify \((r - 5)\) terms

For \((r - 5)\) terms: \((r - 5)\div(r - 5)^3=(r - 5)^{1-3}=(r - 5)^{-2}=\frac{1}{(r - 5)^2}\) (or directly subtract exponents in the denominator: \((r - 5)^{3 - 1}=(r - 5)^2\) in the denominator)

Step6: Combine all simplified parts

Multiply the simplified coefficient, \(p\) term, and divide by simplified \(q\) and \((r - 5)\) terms:
\(\frac{37}{50}\cdot p\cdot\frac{1}{q^3}\cdot\frac{1}{(r - 5)^2}=\frac{37p}{50q^3(r - 5)^2}\)

Answer:

\(\boldsymbol{\frac{37p}{50q^3(r - 5)^2}}\) (the third option: \(\frac{37p}{50q^3(r - 5)^2}\))