Sure! Let's simplify the expression [tex]\(\frac{3^{-6}}{3^{-4}}\)[/tex].
1. Identifying the rule: When dividing exponential terms that have the same base, we keep the base the same and subtract the exponent of the denominator from the exponent of the numerator. The general rule is:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
2. Applying the rule: For the given expression, [tex]\(\frac{3^{-6}}{3^{-4}}\)[/tex], the base is [tex]\(3\)[/tex], the exponent of the numerator is [tex]\(-6\)[/tex], and the exponent of the denominator is [tex]\(-4\)[/tex]. According to the rule, we subtract the exponents.
[tex]\[
3^{-6 - (-4)} = 3^{-6 + 4}
\][/tex]
3. Performing the subtraction: Simplify the exponent:
[tex]\[
-6 + 4 = -2
\][/tex]
4. Simplified expression: The expression simplifies to:
[tex]\[
3^{-2}
\][/tex]
5. Final result as a decimal: We can also convert [tex]\(3^{-2}\)[/tex] to a decimal value. Recall that [tex]\(3^{-2} = \frac{1}{3^2}\)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
\frac{1}{9} = 0.1111111111111111
\][/tex]
So, the simplified expression [tex]\(\frac{3^{-6}}{3^{-4}}\)[/tex] leads to the answer [tex]\(3^{-2}\)[/tex]. Numerically, [tex]\(3^{-2}\)[/tex] equals approximately [tex]\(0.1111111111111111\)[/tex].
