Sure, let's simplify the given expression step-by-step:
The expression given is [tex]\((3n)^{-3}\)[/tex].
1. Understanding Negative Exponents:
A negative exponent means we take the reciprocal of the base and then apply the exponent as a positive number. So, [tex]\((a)^{-b} = \frac{1}{a^b}\)[/tex].
Thus, [tex]\((3n)^{-3} = \frac{1}{(3n)^3}\)[/tex].
2. Applying the Exponent:
Now we need to apply the exponent to the entire base:
[tex]\((3n)^3 = 3^3 \cdot n^3\)[/tex].
3. Calculating Powers:
Calculate the power of 3:
[tex]\(3^3 = 3 \times 3 \times 3 = 27\)[/tex].
Thus, [tex]\((3n)^3 = 27n^3\)[/tex].
4. Forming the Reciprocals:
Now put it all together:
[tex]\((3n)^{-3} = \frac{1}{(3n)^3} = \frac{1}{27n^3}\)[/tex].
So, the simplified form of [tex]\((3n)^{-3}\)[/tex] is:
[tex]\[
\frac{1}{27n^3}
\][/tex]
This is the final answer.
