Question

$\left(\frac{4p^{3}r^{5}}{5wrp}\
ight)^{3}$

Explanation:

Step1: Simplify the fraction inside the parentheses by subtracting exponents of like bases.

For \( p \): \( p^3 \div p^1 = p^{3 - 1}=p^2 \)
For \( r \): \( r^5 \div r^1 = r^{5 - 1}=r^4 \)
So the fraction becomes \( \frac{4p^2r^4}{5w} \)

Step2: Apply the power of a quotient rule \((\frac{a}{b})^n=\frac{a^n}{b^n}\) and power of a product rule \((ab)^n=a^nb^n\).

Raise numerator and denominator to the power of 3:
\( \frac{(4p^2r^4)^3}{(5w)^3} \)
Expand the numerator: \( 4^3(p^2)^3(r^4)^3 = 64p^{6}r^{12} \)
Expand the denominator: \( 5^3w^3 = 125w^3 \)

Answer:

\(\frac{64p^{6}r^{12}}{125w^3}\)