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<h3>Solution and Explanation:</h3>
If and
are directly proportional, then they must have a common scalar.
To find we must first find its scalar. If we let
be the scalar we can form the equation,
.
We are given the information that if then
. We can use that in finding the scalar.
Now we can solve for when
knowing that our scalar is
.
when
Lets start by factoring the 216 into its smaller parts.
![\sqrt[3]{2 * 2 * 2 * 3 * 3 * 3}](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B2%20%2A%202%20%2A%202%20%2A%203%20%2A%203%20%2A%203%7D)
From here, we can separate the three 2s and the three 3s into two separate radicals.
![\sqrt[3]{2*2*2} * \sqrt[3]{3*3*3}](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B2%2A2%2A2%7D%20%2A%20%20%5Csqrt%5B3%5D%7B3%2A3%2A3%7D%20)
Since we have three copies of the same number in each, the answer to the cube root is the number we have the copies of.
![\sqrt[3]{2*2*2} * \sqrt[3]{3*3*3} = 2 * 3](https://tex.z-dn.net/?f=%20%5Csqrt%5B3%5D%7B2%2A2%2A2%7D%20%2A%20%20%5Csqrt%5B3%5D%7B3%2A3%2A3%7D%20%3D%202%20%2A%203)
Finally, we just need to multiply out what remains to find the solution.
So, the final answer is 6.
From here, we can separate the three 2s and the three 3s into two separate radicals.
Since we have three copies of the same number in each, the answer to the cube root is the number we have the copies of.
Finally, we just need to multiply out what remains to find the solution.
So, the final answer is 6.