Certainly! Let's solve the expression step-by-step:
Given expression: [tex]\(\frac{(-3)^{-4}}{(-3)^{-8}}\)[/tex]
1. Applying the properties of exponents:
- When we have the same base in the numerator and the denominator, we can subtract the exponent in the denominator from the exponent in the numerator.
- Mathematically, [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
2. Subtract the exponents:
- Here, the base is [tex]\(-3\)[/tex], the exponent in the numerator is [tex]\(-4\)[/tex], and the exponent in the denominator is [tex]\(-8\)[/tex].
- So, the expression simplifies to [tex]\( (-3)^{(-4) - (-8)} \)[/tex].
3. Simplify the exponent:
- Subtract [tex]\(-8\)[/tex] (which is equivalent to adding 8) from [tex]\(-4\)[/tex]:
[tex]\[
(-4) - (-8) = -4 + 8 = 4
\][/tex]
4. Substitute the simplified exponent:
- Now we have [tex]\( (-3)^4 \)[/tex].
5. Calculate the result:
- To find [tex]\( (-3)^4 \)[/tex], we multiply [tex]\(-3\)[/tex] by itself 4 times:
[tex]\[
(-3) \times (-3) \times (-3) \times (-3)
\][/tex]
- Let's break it down:
[tex]\[
(-3) \times (-3) = 9
\][/tex]
[tex]\[
9 \times (-3) = -27
\][/tex]
[tex]\[
-27 \times (-3) = 81
\][/tex]
So, the value of [tex]\(\frac{(-3)^{-4}}{(-3)^{-8}}\)[/tex] is [tex]\(81\)[/tex].
