Lesson: Chapter - 4

Roots and Radicals

We just saw that roots express fractional exponents. But it is often easier to work with roots in a different format. When a number or term is raised to a fractional power, the expression can be converted into one involving a root in the following way:

with the v sign as the radical sign, and x^a the radicand.

Roots are like exponents, only backward. For example, to square the number 3 is to multiply 3 by itself: 3² = 3 × 3 = 9. The root of 9, &radic 9 is 3. In other words, the square root of a number is the number that, when squared, is equal to the given number.

Square roots are the most commonly used roots, but there are also cube roots (numbers raised to ¹/₃), fourth roots, fifth roots, etc. Each root is represented by a radical sign with the appropriate number next to it (a radical without any superscript denotes a square root). For example, cube roots are shown as ³ &radic fourth roots as ⁴ &radic , and so on. These roots of higher degrees operate the same way square roots do. Because 3³ = 27, it follows that the cube root of 27 is 3.

Here are a few examples:

The same rules that apply to multiplying and dividing exponential terms with the same exponent apply to roots as well. Look for yourself:

Just be sure that the roots are of the same degree (i.e., you are multiplying or dividing all square roots or all roots of the fifth power).

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Mathematics Practice Questions