Let's tackle these problems step-by-step.
### Problem 1: Find the value of [tex]\(\frac{x}{y}\)[/tex]
#### Given:
[tex]\[
\left(\frac{9}{4}\right)^{-10} \times \left(\frac{7}{18}\right)^{-10} = \left(\frac{x}{y}\right)^{-10}
\][/tex]
To solve for [tex]\(\frac{x}{y}\)[/tex], let's simplify the given expression:
1. Rewrite the left-hand side:
[tex]\[
\left(\frac{9}{4}\right)^{-10} \times \left(\frac{7}{18}\right)^{-10}
\][/tex]
2. Using the property of exponents [tex]\((a \times b)^c = a^c \times b^c\)[/tex]:
[tex]\[
\left[(\frac{9}{4}) \times (\frac{7}{18})\right]^{-10}
\][/tex]
3. Simplify the fraction inside:
[tex]\[
\frac{9}{4} \times \frac{7}{18} = \frac{9 \times 7}{4 \times 18} = \frac{63}{72} = \frac{7}{8}
\][/tex]
4. Substitute back into the equation:
[tex]\[
\left(\frac{7}{8}\right)^{-10} = \left(\frac{x}{y}\right)^{-10}
\][/tex]
Since the bases are the same, we equate them:
[tex]\[
\frac{x}{y} = \frac{7}{8}
\][/tex]
Thus, the correct answer is [tex]\(\boxed{2}\)[/tex].
### Problem 2: Find the value of [tex]\(x\)[/tex]
#### Given:
[tex]\[
\left(\frac{-4}{7}\right)^{-5} \times \left(\frac{-4}{7}\right)^x = \left(\frac{-4}{7}\right)^{-3}
\][/tex]
To solve for [tex]\(x\)[/tex], let's use properties of exponents:
1. Combine the exponents on the left-hand side:
[tex]\[
\left(\frac{-4}{7}\right)^{-5 + x} = \left(\frac{-4}{7}\right)^{-3}
\][/tex]
2. Since the bases are the same, equate the exponents:
[tex]\[
-5 + x = -3
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = -3 + 5
\][/tex]
[tex]\[
x = 2
\][/tex]
Thus, the correct answer is [tex]\(x = \boxed{2}\)[/tex].
