To simplify the expression [tex]\(\frac{3^{-6}}{3^{-4}}\)[/tex], you follow these steps:
1. Identify the Operation on Exponents:
- The base is the same in both parts of the fraction, which is [tex]\(3\)[/tex].
2. Subtract the Exponents:
- When dividing powers of the same base, you subtract the exponent of the denominator from the exponent of the numerator.
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
3. Apply the Rule:
- Here, the exponents are [tex]\(-6\)[/tex] and [tex]\(-4\)[/tex]. So, you subtract [tex]\(-4\)[/tex] from [tex]\(-6\)[/tex]:
[tex]\[
-6 - (-4) = -6 + 4 = -2
\][/tex]
4. Simplify the Expression:
- Therefore, the given expression simplifies to:
[tex]\[
\frac{3^{-6}}{3^{-4}} = 3^{-2}
\][/tex]
5. Optional - Convert the Negative Exponent to Positive:
- To express [tex]\(3^{-2}\)[/tex] in a more conventional form, recognize that a negative exponent indicates the reciprocal:
[tex]\[
3^{-2} = \frac{1}{3^2} = \frac{1}{9}
\][/tex]
Thus, the simplified expression is [tex]\(3^{-2}\)[/tex], which is equivalent to [tex]\(\frac{1}{9}\)[/tex] or approximately [tex]\(0.1111111111111111\)[/tex].
