To determine what number should be multiplied by [tex]\(3^{-8}\)[/tex] to get [tex]\(3^{-4}\)[/tex], we can use the properties of exponents.
1. Let's denote the unknown number as [tex]\(x\)[/tex].
2. According to the problem, we need to find [tex]\(x\)[/tex] such that:
[tex]\[
(3)^{-8} \times x = (3)^{-4}
\][/tex]
3. To isolate [tex]\(x\)[/tex], we can divide both sides of the equation by [tex]\(3^{-8}\)[/tex]:
[tex]\[
x = \frac{(3)^{-4}}{(3)^{-8}}
\][/tex]
4. Using the property of exponents which states that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we can simplify the right-hand side of the equation:
[tex]\[
x = 3^{-4 - (-8)} = 3^{-4 + 8} = 3^{4}
\][/tex]
5. Next, we need to calculate [tex]\(3^{4}\)[/tex]:
[tex]\[
3^{4} = 3 \times 3 \times 3 \times 3 = 81
\][/tex]
Therefore, the number by which [tex]\(3^{-8}\)[/tex] should be multiplied to get [tex]\(3^{-4}\)[/tex] is [tex]\(81\)[/tex].
