Evaluate the expression $\left(\frac{2^2 x^2}{x y^3}\right)^2$ for $x=4$ and $y=2$.

Question

ajcosplay6 ajcosplay6

05-11-2026
Mathematics

Answered

Evaluate the expression $\left(\frac{2^2 x^2}{x y^3}\right)^2$ for $x=4$ and $y=2$.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T15:52:06-05:00

Simplify the expression inside the parentheses to $\frac{4x}{y^3}$.
Substitute $x=4$ and $y=2$ into the expression, which yields $\frac{16}{8} = 2$.
Square the resulting value, $2^2$, to find the final result.
The final answer is $\boxed{4}$.

Explanation

Understanding the Problem

We are asked to evaluate the expression $\left(\frac{2^2 x^2}{x y^3}\right)^2$ given that $x = 4$ and $y = 2$. The expression involves constants, variables, exponents, and division. Our approach is to first simplify the algebraic expression inside the parentheses, then substitute the numerical values of $x$ and $y$, and finally compute the result by applying the outer exponent.

Simplifying the Expression

First, let's simplify the inner expression:
\[\frac{2^2 x^2}{x y^3} = \frac{4 x^2}{x y^3}.\]
Since $x^2 / x = x$, we can further simplify this to:
\[\frac{4 x}{y^3}.\]
This makes the entire expression $\left(\frac{4x}{y^3}\right)^2$.

Evaluating and Concluding

Now, we substitute $x = 4$ and $y = 2$ into our simplified expression $\frac{4x}{y^3}$:
\[\frac{4(4)}{2^3} = \frac{16}{8} = 2.\]
Now we apply the square from the original problem, which is $(2)^2$:
\[2^2 = 4.\]
Thus, the final value of the expression for the given variables is 4.

Examples

This type of expression simplification is common in physics and engineering formulas where you might need to determine the final output of a system based on variable inputs. For example, if you are calculating the kinetic energy of an object where velocity (v) is a function of time, you would plug in specific values for time and parameters to find the energy at a specific moment. Mastering these algebraic manipulations ensures accuracy in complex multi-step calculations.