To solve the expression [tex]\(\frac{6^3 \times 6^{-4}}{6^2}\)[/tex], we will break it down step-by-step:
1. Combine the Exponents in the Numerator:
The numerator of the given expression is [tex]\(6^3 \times 6^{-4}\)[/tex]. When multiplying powers of the same base, you add the exponents:
[tex]\[
6^3 \times 6^{-4} = 6^{3 + (-4)} = 6^{-1}
\][/tex]
2. Rewrite the Expression with the Simplified Numerator:
Now, we substitute [tex]\(6^{-1}\)[/tex] into the original expression, giving us:
[tex]\[
\frac{6^{-1}}{6^2}
\][/tex]
3. Simplify the Division of Exponents:
When dividing powers of the same base, you subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
\frac{6^{-1}}{6^2} = 6^{-1 - 2} = 6^{-3}
\][/tex]
4. Rewrite the Negative Exponent as a Positive Exponent:
Recall the property of exponents that [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]. Therefore:
[tex]\[
6^{-3} = \frac{1}{6^3}
\][/tex]
Now we calculate [tex]\(6^3\)[/tex]:
[tex]\[
6^3 = 6 \times 6 \times 6 = 216
\][/tex]
Substituting back in, we get:
[tex]\[
6^{-3} = \frac{1}{216}
\][/tex]
So, the final simplified result of the expression [tex]\(\frac{6^3 \times 6^{-4}}{6^2}\)[/tex] is:
[tex]\[
\frac{1}{216} \approx 0.004629629629629629
\][/tex]
