Sure, let's simplify the expression [tex]\( 5 x \cdot \frac{1}{x^{-7}} \cdot x^{-2} \)[/tex] step-by-step.
1. Initial Expression:
[tex]\[
5 x \cdot \frac{1}{x^{-7}} \cdot x^{-2}
\][/tex]
2. Handling the Fraction:
Recall that [tex]\(\frac{1}{x^{-7}}\)[/tex] can be rewritten using the property of exponents [tex]\(\frac{1}{x^{-a}} = x^a\)[/tex]. Therefore,
[tex]\[
\frac{1}{x^{-7}} = x^{7}
\][/tex]
3. Substitute and Combine:
Now substitute [tex]\(x^7\)[/tex] back into the original expression:
[tex]\[
5 x \cdot x^7 \cdot x^{-2}
\][/tex]
4. Combining Exponents:
Use the property of exponents [tex]\(x^a \cdot x^b = x^{a+b}\)[/tex]:
- Combine the exponents [tex]\(1\)[/tex] (from [tex]\(5x\)[/tex]), [tex]\(7\)[/tex], and [tex]\(-2\)[/tex]:
[tex]\[
5 x^{1 + 7 - 2}
\][/tex]
Simplifying the exponent:
[tex]\[
5 x^{6}
\][/tex]
5. Final Simplified Expression:
[tex]\[
5 x^{6}
\][/tex]
Therefore, the simplified expression is [tex]\(5 x^6\)[/tex].
