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Multiply and simplify completely, including the elimination of all negative exponents:
[tex]\[ \left(\frac{3 x^4}{2 y^7}\right)^4 \cdot \left(\frac{2^3 y^3}{3^2 x^{-4}}\right) \][/tex]


Sagot :

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]