Overview:
Multiplication and division of radicals use the same properties as other types of multiplication. It is much easier to simplify expressions before they are used in an expression. Also, the result needs to be checked carefully to make sure the expression is in its simplest form.
Factoring a Radical
Multiplying a radical is defined algebraically as √ab = √a√b. This definition is very helpful when multiplying two radicals. For example, suppose the problem is -(√6)∙(√3). That is equal to √18, which can further be simplified to 3√2. There are actually two ways to do this, and both arrive at the same answer. One way is to think of √6 as being factored further as √2√3 and multiply by √3. Now, √3 times itself by definition equals 3, and then multiply by √2, so the simplest form of√18 is 3√2. The other way is to think of √18 as √9√2 and then solve as 3√2. The first way illustrates the concept of what a square root means, that √a ∙√a is the same thing as a.
Using the Commutative and Associative Properties
Suppose the number is -3√6 ∙ 4√3. It can be solved by moving the factors around and associating them, as all the operations to be performed are multiplication. This rewrites the problem to (-3∙ 4)(√6√3) or -12(√18), which can be further simplified to -12 ∙3√2 or -36√2.
Division of Radicals
The definition of division of radicals is similar to the definition of multiplication of radicals. If a and b are real numbers, neither are negative, and b is not equal to zero, then √(a/b) is the same thing as √a/√b. It can also be shown by using real numbers as examples. For example, √(9/25) is simplified completely as 3/5. Similarly, √9/√25 is also simplified as 3/5.
Rationalizing the Denominator
In order for the numerator to be in simplest terms, all perfect squares should be factored out from under the radical. In addition, the denominator of the fraction should not contain any radicals either. Suppose the final expression was √2/√5. In order to eliminate the √5 in the denominator, multiply the expression by √5/√5. The numerator will be √2√5 and the denominator of the fraction will be 5. The resulting fraction (√2√5)/5 is in simplest terms.
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