To solve the given expression [tex]\(\frac{(-3)^{-4}}{(-3)^{-8}}\)[/tex], we can use the properties of exponents. Here is a step-by-step solution:
1. Identify the property of exponents to use:
We need to simplify the expression [tex]\(\frac{a^m}{a^n}\)[/tex]. According to the rules of exponents, [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
2. Set up the problem using the rule:
In this case, the base [tex]\(a\)[/tex] is [tex]\(-3\)[/tex], [tex]\(m\)[/tex] is [tex]\(-4\)[/tex], and [tex]\(n\)[/tex] is [tex]\(-8\)[/tex]. Using the property of exponents, we get:
[tex]\[
\frac{(-3)^{-4}}{(-3)^{-8}} = (-3)^{-4 - (-8)}
\][/tex]
3. Simplify the exponent:
Subtract the exponents in the numerator and the denominator:
[tex]\[
-4 - (-8) = -4 + 8 = 4
\][/tex]
4. Rewrite the expression with the simplified exponent:
Now we have:
[tex]\[
(-3)^{4}
\][/tex]
5. Evaluate the exponent:
Calculate [tex]\((-3)^4\)[/tex]:
[tex]\[
(-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 9 \times 9 = 81
\][/tex]
So, the final answer for the expression [tex]\(\frac{(-3)^{-4}}{(-3)^{-8}}\)[/tex] is [tex]\(81\)[/tex].
