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To simplify the sum (or difference) of radicals such as 72 + 32, we proceed just as when combining like terms. Thus,

7x + 3x = (7 + 3)x = 10x

and

72 + 32 = (7 + 3)2 = 102

Similarly,

10 3-4 3+5+2 5 = (10-4) 3+(1+2) 5 = 6 3+3 5

75+ 27=25·3+9·3=5(3) + 33=83

The procedure is the same if the radical contains a variable.

16 x + 4 x= 16·x +4· x = 4x + 2x = 6x

To find the product of radicals, we proceed just as in multiplying polynomials, as the following examples illustrate.

5(x + y) = 5x + 5y

2 [[7+3]]=2 7+2 3=14+6

And, with binomials,

(x + 5)(x - 3) = x2-3 x+5 x-15=x2+2 x-15

(2+5) (2-3) =2 2-3 2+5 2-15=2+22-15= 13+2 2

(Note:  2 2 = 4=2. In general, a a=a if a is positive.)

Examples

Find the following products and simplify.

1. 7 (7-14) = 7 7-7 14=7-98 = 7-49·2= 7-49 2=7-7 2

2. (2+4) (2-4) = (2)2-42 = 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, 53, 78, 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.

53 = 5·33·3 = 5 33

Multiply the numerator and the denominator by 3 because 3·3 gives a rational number.

4x = 4·xx·x = 4 xx

Multiply the numerator and the denominator by x because x·x=x. We have no guarantee that x is rational, but the radical sign does not appear in the denominator of the expression.

78 = 7·28·2= 144

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.

78 = 7·88·8 = 568 = 4·148=2 148 = 144

Before trying to rationalize the denominator for 23-2, recall that the product (a+b) (a-b) results in the difference of two squares:

(a+b) (a-b)=a2-b2

As long as a and b are real numbers, a+b and a-b 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)32-(2)2

The denominator is the difference of two squares.

= 2 (3+2)9-2

The denominator is a rational number.

= 2 (3+2)7

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