If `x=(3/2)^2xx(2/3)^(-4),y=(5/2)^2xx9/25`, then find the value of `x^(-2)y`.

Options

A. `(2/3)^(!0)`
B. `(3/2)^(-12)`
C. `(2/3)^6`
D. 1

Answer

To solve the problem, we need to find the value of \( x^{-2}y \) given:

\[ x = \left(\frac{3}{2}\right)^2 \times \left(\frac{2}{3}\right)^{-4} \]
\[ y = \left(\frac{5}{2}\right)^2 \times \frac{9}{25} \]

### Step 1: Simplify \( x \)

First, we simplify \( x \):

\[ x = \left(\frac{3}{2}\right)^2 \times \left(\frac{2}{3}\right)^{-4} \]

Using the property of exponents, \( a^{-m} = \frac{1}{a^m} \):

\[ \left(\frac{2}{3}\right)^{-4} = \left(\frac{3}{2}\right)^4 \]

Now substituting back into the equation for \( x \):

\[ x = \left(\frac{3}{2}\right)^2 \times \left(\frac{3}{2}\right)^4 \]

Using the property of exponents \( a^m \times a^n = a^{m+n} \):

\[ x = \left(\frac{3}{2}\right)^{2+4} = \left(\frac{3}{2}\right)^6 \]

### Step 2: Simplify \( y \)

Now we simplify \( y \):

\[ y = \left(\frac{5}{2}\right)^2 \times \frac{9}{25} \]

We can rewrite \( 9 \) as \( 3^2 \) and \( 25 \) as \( 5^2 \):

\[ y = \left(\frac{5}{2}\right)^2 \times \frac{3^2}{5^2} \]

This can be rewritten as:

\[ y = \frac{5^2}{2^2} \times \frac{3^2}{5^2} = \frac{3^2}{2^2} = \left(\frac{3}{2}\right)^2 \]

### Step 3: Calculate \( x^{-2}y \)

Now we need to find \( x^{-2}y \):

\[ x^{-2} = \left(\left(\frac{3}{2}\right)^6\right)^{-2} = \left(\frac{3}{2}\right)^{-12} \]

So,

\[ x^{-2}y = \left(\frac{3}{2}\right)^{-12} \times \left(\frac{3}{2}\right)^2 \]

Using the property of exponents again:

\[ x^{-2}y = \left(\frac{3}{2}\right)^{-12 + 2} = \left(\frac{3}{2}\right)^{-10} \]

### Step 4: Final Result

Now we can express this in a more simplified form:

\[ x^{-2}y = \left(\frac{3}{2}\right)^{-10} = \left(\frac{2}{3}\right)^{10} \]

### Conclusion

Thus, the final answer is:

\[ \left(\frac{2}{3}\right)^{10} \]