Areas of figures are often related to two lengths multiplied together, and volumes of objects are often related to three lengths multiplied together. If the lengths are related, this can result in an equation with powers of 2 or 3. Here are some examples:

• The area of a circle with radius is .
• The area of an equilateral triangle with side length is .
• The volume of a sphere with radius is .
• The volume of a regular tetrahedron (a pyramid made from 4 equilateral triangles) with side length is .

We can solve each of these equations for or to give us an equation that would allow us to find the radius or side length from the area or volume. These equations would include either a square root or cube root.

Whenever we have an equation with a radical symbol that contains a variable, we can solve it by isolating the radical and then raising each side of the equation to a power in order to get a new equation without radicals. Here is an example:

Sometimes this results in an equation with solutions that do not make the original equation true. If we use this strategy, it is good to check the solutions to the new equation after raising each side to a power, to be sure they make the original equation true. In this example, we did find a solution to the original equation because .

Another way to solve these equations is to reason about what the answer is, instead of raising each side to a power. For example, if we are solving , we can rearrange it to get and then think, “If the positive square root of is 6, then must be 36, because the positive square root of 36 is 6. So must be , because  .” If we check this result, we see that  is a solution to the original equation because .