Sure, let’s solve this problem step-by-step:
First, let's look at the given problem:
[tex]$
\left( \frac{3 x^4}{2 y^7} \right)^4 \cdot \left( \frac{2^3 y^3}{3^2 x^{-4}} \right)
$[/tex]
### Step 1: Simplify the first expression inside the parentheses
Raise each term inside the first fraction to the power of 4:
[tex]\[
\left( \frac{3 x^4}{2 y^7} \right)^4 = \frac{(3 x^4)^4}{(2 y^7)^4}
\][/tex]
Expanding the terms:
[tex]\[
(3 x^4)^4 = 3^4 \cdot x^{4 \cdot 4} = 81 x^{16}
\][/tex]
[tex]\[
(2 y^7)^4 = 2^4 \cdot y^{7 \cdot 4} = 16 y^{28}
\][/tex]
Putting these back into the fraction:
[tex]\[
\frac{81 x^{16}}{16 y^{28}}
\][/tex]
### Step 2: Rewrite the second expression inside the parentheses
It's already simplified, but we can rewrite it clearly:
[tex]\[
\frac{2^3 y^3}{3^2 x^{-4}} = \frac{8 y^3}{9 x^{-4}}
\][/tex]
### Step 3: Now, multiply these two fractions
Combine the two fractions:
[tex]\[
\left( \frac{81 x^{16}}{16 y^{28}} \right) \cdot \left( \frac{8 y^3}{9 x^{-4}} \right)
\][/tex]
Multiply the numerators and the denominators:
[tex]\[
\frac{81 x^{16} \cdot 8 y^3}{16 y^{28} \cdot 9 x^{-4}}
\][/tex]
### Step 4: Simplify the result
Combine the constants:
[tex]\[
81 \cdot 8 = 648
\][/tex]
[tex]\[
16 \cdot 9 = 144
\][/tex]
So, we now have:
[tex]\[
\frac{648 \cdot x^{16} \cdot y^3}{144 \cdot y^{28} \cdot x^{-4}}
\][/tex]
Simplify the constant fraction:
[tex]\[
\frac{648}{144} = 4.5 = \frac{9}{2}
\][/tex]
Simplify the variables by subtracting powers (using the properties of exponents):
For [tex]\( x \)[/tex]:
[tex]\[
x^{16} \cdot x^4 = x^{16+4} = x^{20}
\][/tex]
For [tex]\( y \)[/tex]:
[tex]\[
y^3 \cdot y^{-28} = y^{3-28} = y^{-25}
\][/tex]
Putting it all together:
[tex]\[
\frac{9 x^{20}}{2 y^{25}}
\][/tex]
Since [tex]\( y^{25} \)[/tex] is in the denominator and has a positive exponent, the expression is now completely simplified.
### Final Answer:
[tex]\[
\frac{9 x^{20}}{2 y^{25}}
\][/tex]
