SOLUTION: Here is an application problem from a chapter on radicals. Surface area of a cube. The formula A=6V^2/3 gives the surface area of a cube in terms of its volume V. What is the vo

Question 277174: Here is an application problem from a chapter on radicals.
Surface area of a cube. The formula A=6V^2/3 gives the surface area of a cube in terms of its volume V. What is the volume of a cube with surface area 12 square feet?
Thanks.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

You're given A so:

We're solving for V so we need to get rid of the 6 in front and the exponent of 2/3. We can eliminate the 6 by dividing both sides by 6:

Now how do we get rid of the exponent? Well, to be perfectly correct, we are not getting rid of the exponent we are changing it to a 1 (since . So how do we change the exponent to a 1?

One way is to raise each side to the 3/2 power. (If you don't see why this works yet, you will shortly.)

On the right side, the rule is to multiply the exponents. 2/3 and 3/2 are reciprocals of each other. And what do you get when you multiply reciprocals? Answer 1! This is why I raised both sides to the 3/2 power:

I knew that raising a power to a power would result in multiplying exponents
I knew that I wanted an exponent of 1
I knew that multiplying reciprocals always results in 1
3/2 is the reciprocal of 2/3

Now we only have to simplify the left side. If you are not yet comfortable with fractional exponents, it can be helpful to look at the exponent in factored form. The factor of three in the exponent says we'll be cubing something and the factor of 1/2 in the exponent says there will be a square root. And, since multiplication is Commutative, we can do these things in any order we choose. Since 2 is not a prefect square, doing the square root first has little appeal. So I'll start by cubing 2. 2 cubed is 8 and then we apply the square root giving us . So:

The only thing left is to simplify the square root. There is a perfect square factor in 8:

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