To simplify the expression [tex]\(\frac{3^{-6}}{3^{-4}}\)[/tex], we need to use the properties of exponents. Here we go step-by-step:
1. Identify the rule: The expression involves division of exponents with the same base. The rule we use is [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
2. Apply the rule: For our specific problem, we have:
[tex]\[
\frac{3^{-6}}{3^{-4}} = 3^{(-6) - (-4)}
\][/tex]
3. Simplify the exponent: Subtracting [tex]\(-4\)[/tex] is the same as adding 4. So, we have:
[tex]\[
(-6) - (-4) = -6 + 4 = -2
\][/tex]
4. Write the simplified expression: With the simplified exponent, the expression now becomes:
[tex]\[
3^{-2}
\][/tex]
5. Evaluate the expression: To find the numerical value of [tex]\(3^{-2}\)[/tex], we know that a negative exponent indicates the reciprocal:
[tex]\[
3^{-2} = \frac{1}{3^2} = \frac{1}{9}
\][/tex]
Therefore, the simplified expression [tex]\(\frac{3^{-6}}{3^{-4}}\)[/tex] results in [tex]\(3^{-2}\)[/tex], which equals [tex]\(\frac{1}{9}\)[/tex].
In summary:
[tex]\[
\frac{3^{-6}}{3^{-4}} = 3^{-2} = \frac{1}{9}
\][/tex]
