To solve the given problem, we need to evaluate both sides of the equation and show that they are equal. The equation is:
\[
\left( \frac{3}{4} \right)^{-3} = \left( \frac{4}{3} \right)^3
\]
Step 1: Simplify the left-hand side (LHS)
The left-hand side is \(\left( \frac{3}{4} \right)^{-3}\).
Using the property of exponents that \(a^{-n} = \frac{1}{a^n}\), we can rewrite this as:
\[
\left( \frac{3}{4} \right)^{-3} = \frac{1}{\left( \frac{3}{4} \right)^3}
\]
Next, we need to compute \(\left( \frac{3}{4} \right)^3\):
\[
\left( \frac{3}{4} \right)^3 = \frac{3^3}{4^3} = \frac{27}{64}
\]
Thus,
\[
\left( \frac{3}{4} \right)^{-3} = \frac{1}{\frac{27}{64}} = \frac{64}{27}
\]
So, the left-hand side simplifies to:
\[
\left( \frac{3}{4} \right)^{-3} = \frac{64}{27}
\]
Step 2: Simplify the right-hand side (RHS)
The right-hand side is \(\left( \frac{4}{3} \right)^3\).
We compute this directly:
\[
\left( \frac{4}{3} \right)^3 = \frac{4^3}{3^3} = \frac{64}{27}
\]
So, the right-hand side simplifies to:
\[
\left( \frac{4}{3} \right)^3 = \frac{64}{27}
\]
Step 3: Compare both sides
From the calculations above, we have:
\[
\left( \frac{3}{4} \right)^{-3} = \frac{64}{27}
\]
\[
\left( \frac{4}{3} \right)^3 = \frac{64}{27}
\]
Since both sides are equal, the equation is verified.
Final Answer:
\[
\boxed{\frac{64}{27}}
\]
Parent Tip: Review the logic above to help your child master the concept of adding and subtracting exponents.
