Answer :

To simplify the expression [tex]\( x^{-6} \cdot x^9 \cdot x^{-4} \)[/tex], we can use the laws of exponents. The laws of exponents tell us that when multiplying powers with the same base, we can add their exponents. Here's a step-by-step solution:

1. Identify the Base: In this case, the base of all expressions is [tex]\( x \)[/tex].

2. Add the Exponents: According to the rule [tex]\( a^m \times a^n = a^{m+n} \)[/tex], we can add the exponents of each [tex]\( x \)[/tex] term:

- Start with the exponents: [tex]\(-6\)[/tex], [tex]\(9\)[/tex], and [tex]\(-4\)[/tex].
- Add them together:
[tex]\[
-6 + 9 - 4
\][/tex]

3. Calculate the Total Exponent:

- First, [tex]\(-6 + 9 = 3\)[/tex].
- Then, [tex]\(3 - 4 = -1\)[/tex].

So, the combined exponent is [tex]\(-1\)[/tex].

4. Rewrite the Expression: The expression with the combined exponent becomes:
[tex]\[
x^{-1}
\][/tex]

5. Express with a Positive Exponent: To write [tex]\( x^{-1} \)[/tex] with a positive exponent, use the rule that [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex]:

[tex]\[
x^{-1} = \frac{1}{x}
\][/tex]

6. Final Answer: The simplified expression, with a positive exponent, is:
[tex]\[
\frac{1}{x}
\][/tex]

So, [tex]\( x^{-6} \cdot x^9 \cdot x^{-4} = \frac{1}{x} \)[/tex].

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Rewritten by : Barada