To simplify the expression [tex]\(5 x \cdot \frac{1}{x^{-7}} \cdot x^{-2}\)[/tex], we will follow these steps:
1. Identify and Rewrite the Expression:
The given expression is:
[tex]\[
5 x \cdot \frac{1}{x^{-7}} \cdot x^{-2}
\][/tex]
2. Simplify the Fraction:
The fraction part [tex]\(\frac{1}{x^{-7}}\)[/tex] can be simplified by remembering that dividing by a negative exponent is equivalent to multiplying by its positive exponent:
[tex]\[
\frac{1}{x^{-7}} = x^{7}
\][/tex]
3. Substitute Back into the Expression:
Substituting back, we get:
[tex]\[
5 x \cdot x^{7} \cdot x^{-2}
\][/tex]
4. Combine the Powers of [tex]\(x\)[/tex]:
When multiplying terms with the same base, add the exponents:
[tex]\[
x \cdot x^7 = x^{1+7} = x^{8}
\][/tex]
and then
[tex]\[
x^{8} \cdot x^{-2} = x^{8-2} = x^6
\][/tex]
5. Combine with the Coefficient:
Now multiply the remaining coefficient:
[tex]\[
5 \cdot x^6 = 5x^6
\][/tex]
Thus, the simplified form of the expression [tex]\(5 x \cdot \frac{1}{x^{-7}} \cdot x^{-2}\)[/tex] is:
[tex]\[
5x^6
\][/tex]
So, the final answer is:
[tex]\[
5x^6
\][/tex]
