Simplify:

[tex]\[
\frac{\left(2 x^{-1} y^{-3}\right)^3}{\left(x^{-2} y^{-5}\right)^{-2}}
\][/tex]

Write your answer using only positive exponents.



Answer :

Sure, let's simplify the given expression step by step.

Given expression:
[tex]\[ \frac{\left(2 x^{-1} y^{-3}\right)^3}{\left(x^{-2} y^{-5}\right)^{-2}} \][/tex]

### Step 1: Simplify the numerator
First, let's simplify the numerator \(\left(2 x^{-1} y^{-3}\right)^3\):
[tex]\[ (2 x^{-1} y^{-3})^3 = 2^3 \cdot (x^{-1})^3 \cdot (y^{-3})^3 = 8 x^{-3} y^{-9} \][/tex]

So the numerator simplifies to:
[tex]\[ 8 x^{-3} y^{-9} \][/tex]

### Step 2: Simplify the denominator
Now, let's simplify the denominator \(\left(x^{-2} y^{-5}\right)^{-2}\):
[tex]\[ (x^{-2} y^{-5})^{-2} = (x^{-2})^{-2} \cdot (y^{-5})^{-2} = x^{4} \cdot y^{10} \][/tex]

So the denominator simplifies to:
[tex]\[ x^4 y^{10} \][/tex]

### Step 3: Simplify the whole fraction
Now, we put the simplified numerator and denominator together and simplify the fraction:
[tex]\[ \frac{8 x^{-3} y^{-9}}{x^4 y^{10}} \][/tex]

This can be simplified by subtracting the exponents of the same bases:
[tex]\[ 8 \cdot x^{-3 - 4} \cdot y^{-9 - 10} = 8 \cdot x^{-7} \cdot y^{-19} \][/tex]

### Step 4: Convert to positive exponents
Finally, we rewrite the expression using only positive exponents:
[tex]\[ 8 \cdot x^{-7} \cdot y^{-19} = \frac{8}{x^{7} y^{19}} \][/tex]

So, the simplified expression using only positive exponents is:
[tex]\[ \frac{8}{x^{7} y^{19}} \][/tex]