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Math Review of Simplifying Radicals

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Before many operations can be performed with expressions involving radicals, the radicals must be simplified.  This ensures that they are factored so that there are no perfect square factors under the radical sign.  Approximations of the nonnegative square root can then be calculated.

How Are Radicals Factored?

The process of factoring radicals is similar to factoring other numbers. For example, a number such as √400 can be simplified to the rational number 20, because 20 times 20 = 400.  It could also be factored as √4√100, because both 4 and 100 are factors of 400.  In this example, the square root of 4 is 2 and the square root of 100 is 10 and 2 ∙ 10 equals 20.  Speaking algebraically, if a and b are real numbers greater than or equal to zero, then √ab = √a√b.

Finding Perfect Squares

Using the algebraic definition, the radical can be broken down into parts that are easier to solve.  For example, the square root of 50 equals √25√2, or 5√2.  Similarly, an expression such as 3√8 can be simplified as (3)√4√2, or 3∙2√2 or 6√2.  The important thing is to find the largest possible perfect square and another factor, then solve.

What About Variables?

The exponents for variables can be factored using the rules of exponents.  Suppose the expression is √4d5.  It can be factored as √4√d5, or 2√d5, but the variable √d5 is not in its simplest terms.  The variable √d5 can be expressed as 2d2√d.

What About Combinations of Variables and Numbers?

To factor expressions that combine more than one variable, each variable can be simplified to find the largest possible perfect square, then combined in steps.  Suppose the expression is √144a9b2.  It can be factored as √144√a9√b2.  The square root of 144 is 12, the square root of a9 is √a8∙√a or a4√a, and the square root of b2 is b.  Putting it all together, the simplified expression is 12ba4√a.

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