Answer :
To find the length of one side of a cube given its volume, we start with the formula for the volume of a cube:
[tex]\[ V = s^3 \][/tex]
where \( V \) is the volume and \( s \) is the side length.
Given that the volume of the cube is \( 27 n^{27} \) cubic units, we need to find \( s \) such that:
[tex]\[ s^3 = 27 n^{27} \][/tex]
To isolate \( s \), we take the cube root of both sides of the equation:
[tex]\[ s = \sqrt[3]{27 n^{27}} \][/tex]
We can simplify this step-by-step:
[tex]\[ 27 = 3^3 \][/tex]
[tex]\[ n^{27} = (n^9)^3 \][/tex]
Thus, combining these observations, the equation becomes:
[tex]\[ s = \sqrt[3]{(3^3 \cdot (n^9)^3)} \][/tex]
Taking the cube root simplifies it to:
[tex]\[ s = 3 \cdot n^9 \][/tex]
Therefore, the length of one side of the cube is:
[tex]\[ 3 n^9 \][/tex]
So, the correct answer is:
[tex]\[ 3 n^9 \text{ units} \][/tex]
[tex]\[ V = s^3 \][/tex]
where \( V \) is the volume and \( s \) is the side length.
Given that the volume of the cube is \( 27 n^{27} \) cubic units, we need to find \( s \) such that:
[tex]\[ s^3 = 27 n^{27} \][/tex]
To isolate \( s \), we take the cube root of both sides of the equation:
[tex]\[ s = \sqrt[3]{27 n^{27}} \][/tex]
We can simplify this step-by-step:
[tex]\[ 27 = 3^3 \][/tex]
[tex]\[ n^{27} = (n^9)^3 \][/tex]
Thus, combining these observations, the equation becomes:
[tex]\[ s = \sqrt[3]{(3^3 \cdot (n^9)^3)} \][/tex]
Taking the cube root simplifies it to:
[tex]\[ s = 3 \cdot n^9 \][/tex]
Therefore, the length of one side of the cube is:
[tex]\[ 3 n^9 \][/tex]
So, the correct answer is:
[tex]\[ 3 n^9 \text{ units} \][/tex]
