Question

question 4

1. which expression is equivalent to the expression below?

\\(\left(\frac{12r^{-12}s^2}{6r^{-4}s^{-2}}\
ight)^3\\)
a. \\(\frac{8s^{12}}{r^{24}}\\) c. \\(\frac{6r^{24}}{s^{12}}\\)
b. \\(8r^9\\) d. \\(6r^{-24}s^{12}\\)
\\(\bigcirc\\) a
\\(\bigcirc\\) b
\\(\bigcirc\\) c
\\(\bigcirc\\) d

Explanation:

Step1: Simplify the coefficient and variables inside the parentheses

First, simplify the coefficient: $\frac{12}{6} = 2$.
For the variable \( r \): Use the rule \( \frac{a^m}{a^n}=a^{m - n} \), so \( r^{-12-(-4)}=r^{-12 + 4}=r^{-8} \).
For the variable \( s \): Use the rule \( \frac{a^m}{a^n}=a^{m - n} \), so \( s^{2-(-2)}=s^{2 + 2}=s^{4} \).
So the expression inside the parentheses becomes \( 2r^{-8}s^{4} \).

Step2: Apply the power of a power rule \((a^m)^n=a^{m\times n}\)

Raise each part to the power of 3:

• Coefficient: \( 2^3=8 \)
• For \( r \): \( (r^{-8})^3=r^{-8\times3}=r^{-24}=\frac{1}{r^{24}} \)
• For \( s \): \( (s^{4})^3=s^{4\times3}=s^{12} \)

Combining these, we get \( \frac{8s^{12}}{r^{24}} \).

Answer:

A. \(\frac{8s^{12}}{r^{24}}\)