Certainly! Let's find the length of one side of the cube given that the volume of the cube is [tex]\( 27 n^{27} \)[/tex] cubic units.
The formula for the volume [tex]\( V \)[/tex] of a cube, where the side length is [tex]\( s \)[/tex], is:
[tex]\[
V = s^3
\][/tex]
Given the volume of the cube is:
[tex]\[
27 n^{27}
\][/tex]
we need to find [tex]\( s \)[/tex] such that:
[tex]\[
s^3 = 27 n^{27}
\][/tex]
To find [tex]\( s \)[/tex], we take the cube root of both sides of the equation:
[tex]\[
s = \sqrt[3]{27 n^{27}}
\][/tex]
Now, we simplify this expression. We know that the cube root of a product is the product of the cube roots:
[tex]\[
\sqrt[3]{27 n^{27}} = \sqrt[3]{27} \cdot \sqrt[3]{n^{27}}
\][/tex]
First, we find the cube root of 27:
[tex]\[
\sqrt[3]{27} = 3
\][/tex]
Next, we find the cube root of [tex]\( n^{27} \)[/tex]:
[tex]\[
\sqrt[3]{n^{27}} = n^{27/3} = n^9
\][/tex]
Combining these results, we have:
[tex]\[
s = 3 \cdot n^9
\][/tex]
So, the length of one side of the cube is:
[tex]\[
3 n^9 \, \text{units}
\][/tex]
Therefore, the correct answer is:
[tex]\[
3 n^9 \, \text{units}
\][/tex]
Hence, the answer is [tex]\( \boxed{3 n^9} \)[/tex] units.
