Sure, let's solve the problem step-by-step with the given values [tex]\(x = 2\)[/tex] and [tex]\(y = 4\)[/tex].
1. First, substitute [tex]\(x\)[/tex] and [tex]\(y\)[/tex] into the first term [tex]\(\left(\frac{x}{y}\right)^{x-y}\)[/tex]:
[tex]\[
\left(\frac{x}{y}\right)^{x-y} = \left(\frac{2}{4}\right)^{2-4}
\][/tex]
2. Calculate the fraction inside the parentheses:
[tex]\[
\frac{2}{4} = \frac{1}{2}
\][/tex]
3. Now calculate the exponent [tex]\(2 - 4\)[/tex]:
[tex]\[
2 - 4 = -2
\][/tex]
4. Therefore, the first term becomes:
[tex]\[
\left(\frac{1}{2}\right)^{-2}
\][/tex]
5. Recall that [tex]\(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}\)[/tex]:
[tex]\[
\left(\frac{1}{2}\right)^{-2} = \left(\frac{2}{1}\right)^{2} = 2^2 = 4
\][/tex]
6. Next, substitute [tex]\(x\)[/tex] and [tex]\(y\)[/tex] into the second term [tex]\(\left(\frac{y}{x}\right)^{y-x}\)[/tex]:
[tex]\[
\left(\frac{y}{x}\right)^{y-x} = \left(\frac{4}{2}\right)^{4-2}
\][/tex]
7. Calculate the fraction inside the parentheses:
[tex]\[
\frac{4}{2} = 2
\][/tex]
8. Now calculate the exponent [tex]\(4 - 2\)[/tex]:
[tex]\[
4 - 2 = 2
\][/tex]
9. Therefore, the second term becomes:
[tex]\[
2^2 = 4
\][/tex]
10. Finally, add the two terms together:
[tex]\[
4 + 4 = 8
\][/tex]
Thus, the solution to the expression is:
[tex]\[
\boxed{8}
\][/tex]
