8.3 Operating With Radical Expressions

Unit: Roots and Radicals

Module: Operations with Radical Expressions

Adding and Subtracting Radical Expressions

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The square root of a sum is not the sum of the square roots. Combine like radicals as like terms.

A very common error must be avoided. In finding roots it is not acceptable to separate an expression into the sum of squares, then declare that the root is the sum of the roots of those squares. Finding roots involves undoing multiplication, not addition. Factor the number under each radical so that at least one of the factors is a perfect square. Simplify the radicals with the perfect square factors. Simplify further by multiplying. Last, combine the like terms. Terms with the same radical are like terms. When the root index divides into the exponent, only the whole number exponent with its base steps out of the radical. Any remainder is left attached to its base under the radical. In the first term in this example, there are only two xs and one y under cuberoot radical. Therefore, no xs or ys can emerge as three are required for each single unit that emerges from the radical. Once each term is simplified as much as possible, notice the radicals are the same in both terms. However, the coefficients cannot be combined because they are not like. The only additional simplification possible is the factoring of the radical.

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1927.doc –rev 03/28/2006

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Unit: Roots and Radicals

Module: Operations with Radical Expressions

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Rationalizing Denominators

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The number 1 can be written as a fraction with any number or expression as the numerator so long as the same number or expression is used as the denominator (except for 0). In other words, 1 = 78/78 or (3a + b)/(3a + b), and so on. Clearing a denominator of radicals means to multiply the fraction by a version of 1. Your version of 1 will be the radical over itself. So, the denominator multiplies out the radical to clear it, and the numerator gets a radical. The conjugate of a given binomial is a binomial that contains the same terms but with the opposite sign connecting the two terms; i.e., (a - b) and (a + b) are conjugates of each other. When you multiply a binomial by its conjugate, you are left with the difference of two squares and no middle terms. FOIL: Multiply binomials by multiplying • First terms together • Outer terms together • Inside terms • Last terms, and adding all the products together This guarantees that you multiply everything and get all your products without losing any. Mathematicians like to eliminate radicals from the denominators of fractions. It’s easy. Multiply it by your version of 1, 3 . 3 Now there is no radical in the denominator.

When the denominator is a binomial, you must be careful. Your version of 1 must use the Conjugate of the denominator. In this case, the conjugate is ( 2 − 1 ).

Remember: You are multiplying binomials and use your FOIL properly so you get the right answer. Now that you’ve cleared the radical from the denominator, solve your equation.

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6853 –rev 05/18/2001

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