Properties of Radicals

An engineer is designing a cube-shaped water tank that can hold up to five hundred cubic meters of water.

To build the tank, he needs the length of each edge. Since the tank is a cube, the edge length is the cube root of five hundred.

Since five hundred is not a perfect cube, he factors it into one hundred twenty-five times four, choosing one hundred twenty-five because its cube root is five.

The nth root of a product equals the product of the nth roots. Applying this gives the cube root of one hundred twenty-five times the cube root of four. Since the cube root of one hundred twenty-five is five, this simplifies to five times the cube root of four.

Before construction, he tests a four-cubic-meter prototype, whose edge length is the cube root of four meters. Comparing the edge lengths shows how measurements from the prototype can be scaled to the original tank.

The nth root of a quotient equals the quotient of the nth roots. Applying this gives the cube root of five hundred over four, which simplifies to the cube root of one hundred twenty-five, which equals five. This means the original tank's edge is five times the prototype's.