


Addition and Multiplication of Radicals & Rationalizing Denominators
Addition and Multiplication of Radicals
7√2 + 3√2 = (7 + 3)√2 = 10√2 Similarly,
To add √75 + √27, we simplify each radical first, then combine like radicals if possible. √75+ √27=√25·3+√9·3=5(√3) + 3√3=8√3The procedure is the same if the radical contains a variable. √16 x + √4 x= √16·√x +√4· √x = 4√x + 2√x = 6√xTo find the product of radicals, we proceed just as in multiplying polynomials, as the following examples illustrate. 5(x + y) = 5x + 5y
And, with binomials, (x + 5)(x  3) = x^{2}3 x+5 x15=x^{2}+2 x15
Examples Find the following products and simplify.
2. (√2+4) (√24) = (√2)^{2}4^{2} = 2  16 = 14
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3. (√5+√3) (√5+√3)=√5 √5+2 √5 √3+√3 √3=5+2 √15+3=8+2 √15
Rationalizing Denominators Each of the expressions, 5√3, √7√8, and 23√2 contains a radical in the denominator that is an irrational number. Such expressions are not considered in simplest form. The objective is to find an equal fraction that has a rational number for a denominator. That is, we want to simplify the expression by rationalizing the denominator. 5√3 = 5·√3√3·√3 = 5 √33Multiply the numerator and the denominator by √3 because √3·√3 gives a rational number. 4√x = 4·√x√x·√x = 4 √xxMultiply the numerator and the denominator by √x Multiply the numerator and denominator by √2 because [√8·√2] = √16 and 16 is a perfect square number. If we had multiplied by √8, the results would have been the same, but the fraction would have to be reduced. √7√8 = √7·√8√8·√8 = √568 = √4·√148=2 √148 = √144Before trying to rationalize the denominator for 23√2, recall that the product (a+b) (ab) results in the difference of two squares: (a+b) (ab)=a^{2}b^{2}As long as a and b are real numbers, a+b and ab are called conjugate surds of each other. Therefore, if the numerator and the denominator of a fraction are multiplied by the conjugate surd of the denominator, the denominator will be the difference of two squares and will be a rational number. 23√2 = 2 (3+√2)(3√2) (3+√2) 3+√2 is the conjugate surd of 3√2.= 2 (3+√2)3^{2}(√2)^{2} The denominator is the difference of two squares. = 2 (3+√2)92 = 2 (3+√2)7
