How to Write Expressions as Radicals

By Grace Williams

Radicals, or roots, are the mathematical opposites of exponents. The smallest root, the square root, is the opposite of squaring a number, so x^2 (or x squared) = √x. The next highest root, the cube root, is equal to raising a number to the third power: x^3 = ³√x. The small 3 above the radical is called an index number, and that number represents the exponent opposite. Because of their relationship, radicals and exponents can be used to cancel each other out or to convert between each other. For example, ³√x equals x^(1/3).

Write the expression (x^2)^(4/3) into radical form. Note that the (x^2) is the base and the (4/3) is its exponent.

Use the base law of exponents, which states that (x^m)^n equals x^(m * n). Multiply the exponent on the base by the other exponent: x^(2 * 4/3) or x^(8/3). Note that the base law also works in the opposite direction and that x^(8/3) equals x^(8 * (1/3)). Pull the 8 out of the exponent to simplify: x^8^(1/3). Note that (1/3) is equivalent to ³√x.

Use the cube root to cancel out the exponent: ³√(x^8). Leave the answer as it is for the radical form.