To simplify the expression [tex]\(\frac{x^{-6}}{x^{-4}}\)[/tex] and express it using positive exponents only, we will follow these steps:
1. Recall the property of exponents:
For any non-zero base [tex]\(a\)[/tex], and any integers [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
2. Apply the property to the numerator and denominator:
Here, our base [tex]\(a\)[/tex] is [tex]\(x\)[/tex]. So, we apply the property to the exponents:
[tex]\[
\frac{x^{-6}}{x^{-4}} = x^{-6 - (-4)}
\][/tex]
3. Simplify the exponent:
We need to simplify the expression in the exponent:
[tex]\[
-6 - (-4) = -6 + 4 = -2
\][/tex]
Thus:
[tex]\[
x^{-6 - (-4)} = x^{-2}
\][/tex]
4. Convert to a positive exponent:
Recall that [tex]\(x^{-a} = \frac{1}{x^a}\)[/tex]. Therefore:
[tex]\[
x^{-2} = \frac{1}{x^2}
\][/tex]
5. Express the final answer:
Since we are asked to express our answer using positive exponents only, we should present the simplified expression:
[tex]\[
\frac{x^{-6}}{x^{-4}} = x^{-2} = \frac{1}{x^2}
\][/tex]
Thus, the simplified expression using positive exponents only is:
[tex]\[
\frac{x^{-6}}{x^{-4}} = x^2
\][/tex]
