To simplify the expression [tex]\(\frac{3^{-6}}{3^{-4}}\)[/tex], follow these steps:
1. Identify the base and the exponents:
In the given expression, the base is 3, the numerator has an exponent of -6, and the denominator has an exponent of -4.
2. Recall the property of exponents for division:
When you divide two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. This property can be written as:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
3. Apply the property to the given expression:
[tex]\[
\frac{3^{-6}}{3^{-4}} = 3^{-6 - (-4)}
\][/tex]
4. Simplify the exponent:
Calculate [tex]\(-6 - (-4)\)[/tex]:
[tex]\[
-6 - (-4) = -6 + 4 = -2
\][/tex]
5. Write the simplified expression:
[tex]\[
\frac{3^{-6}}{3^{-4}} = 3^{-2}
\][/tex]
6. Find the value of the simplified expression:
Raising 3 to the power of -2 means taking the reciprocal of 3 squared. We express this as:
[tex]\[
3^{-2} = \frac{1}{3^2} = \frac{1}{9}
\][/tex]
Therefore, the simplified expression is [tex]\(3^{-2}\)[/tex] and its value is [tex]\(\frac{1}{9}\)[/tex], which is approximately [tex]\(0.1111111111111111\)[/tex].
