Alright, let's simplify the given expression step by step:
[tex]\[
\frac{g^{-7} x^{-6}}{g^{-5} x^{-2}}
\][/tex]
### Step 1: Combine Exponents for the Same Base
We can combine the exponents for each base ([tex]\(g\)[/tex] and [tex]\(x\)[/tex]) by subtracting the exponent in the denominator from the exponent in the numerator.
For [tex]\(g\)[/tex]:
[tex]\[
g^{-7} \div g^{-5} = g^{-7 - (-5)} = g^{-7 + 5} = g^{-2}
\][/tex]
For [tex]\(x\)[/tex]:
[tex]\[
x^{-6} \div x^{-2} = x^{-6 - (-2)} = x^{-6 + 2} = x^{-4}
\][/tex]
Thus, our expression becomes:
[tex]\[
g^{-2} x^{-4}
\][/tex]
### Step 2: Write the Exponents Without Negative Signs
To write the exponents without negative signs, recall that [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]. Therefore:
[tex]\[
g^{-2} = \frac{1}{g^2}
\][/tex]
[tex]\[
x^{-4} = \frac{1}{x^4}
\][/tex]
Putting it together, we get:
[tex]\[
g^{-2} x^{-4} = \frac{1}{g^2} \cdot \frac{1}{x^4}
\][/tex]
### Step 3: Combine the Results into a Single Fraction
[tex]\[
\frac{1}{g^2} \cdot \frac{1}{x^4} = \frac{1}{g^2 x^4}
\][/tex]
### Final Answer
So, the simplified expression, written without negative exponents, is:
[tex]\[
\frac{1}{g^2 x^4}
\][/tex]
