Sure, let's simplify the given expression step-by-step. The given expression is:
[tex]\[
\frac{10 x^{-3} y^{-1}}{8 x^3 y^{-5}}
\][/tex]
### Step 1: Simplify the coefficients
First, we deal with the numerical coefficients:
[tex]\[
\frac{10}{8} = \frac{5}{4}
\][/tex]
### Step 2: Simplify the [tex]\(x\)[/tex] term
Next, let's simplify the [tex]\(x\)[/tex] terms by using the properties of exponents:
[tex]\[
\frac{x^{-3}}{x^3} = x^{-3 - 3} = x^{-6}
\][/tex]
### Step 3: Simplify the [tex]\(y\)[/tex] term
Similarly, simplify the [tex]\(y\)[/tex] terms:
[tex]\[
\frac{y^{-1}}{y^{-5}} = y^{-1 - (-5)} = y^{-1 + 5} = y^4
\][/tex]
### Step 4: Combine the simplified terms
Now, let's combine the results from the previous steps to get the final simplified expression:
[tex]\[
\frac{10 x^{-3} y^{-1}}{8 x^3 y^{-5}} = \frac{5}{4} \cdot x^{-6} \cdot y^4
\][/tex]
### Step 5: Express [tex]\(x^{-6}\)[/tex] with positive exponents
Finally, convert [tex]\(x^{-6}\)[/tex] to a positive exponent:
[tex]\[
x^{-6} = \frac{1}{x^6}
\][/tex]
So the expression becomes:
[tex]\[
\frac{5}{4} \cdot \frac{1}{x^6} \cdot y^4 = \frac{5}{4} \cdot \frac{y^4}{x^6} = \frac{5y^4}{4x^6}
\][/tex]
Thus, the simplified expression with positive exponents is:
[tex]\[
\boxed{\frac{5y^4}{4x^6}}
\][/tex]
