To simplify the expression [tex]\(\frac{3^{-6}}{3^{-4}}\)[/tex], we use the rules of exponents. Specifically, we apply the rule that states [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]. Here is the step-by-step solution:
1. Identify the base and the exponents: The base is 3 in both the numerator and the denominator. The exponent in the numerator is -6, and the exponent in the denominator is -4.
2. Apply the exponent rule for division: According to the exponent rule, when you divide like bases, you subtract the exponent in the denominator from the exponent in the numerator. Therefore, we compute:
[tex]\[
3^{-6 - (-4)}
\][/tex]
3. Simplify the exponent subtraction:
[tex]\[
-6 - (-4) = -6 + 4 = -2
\][/tex]
4. Rewrite the expression with the simplified exponent:
[tex]\[
3^{-2}
\][/tex]
5. Convert the negative exponent to a positive exponent by taking the reciprocal: Recall that [tex]\(a^{-m} = \frac{1}{a^m}\)[/tex]. Thus,
[tex]\[
3^{-2} = \frac{1}{3^2}
\][/tex]
6. Calculate the value of [tex]\(3^2\)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
7. Express the final simplified result:
[tex]\[
3^{-2} = \frac{1}{9}
\][/tex]
Therefore, the simplified expression [tex]\(\frac{3^{-6}}{3^{-4}}\)[/tex] results in [tex]\(\boxed{\frac{1}{9}}\)[/tex].
In other terms, [tex]\(3^{-2} = \frac{1}{9} \approx 0.1111111111\)[/tex].
