Addition and Multiplication of Radicals & Rationalizing Denominators

Addition and Multiplication of Radicals

To simplify the sum (or difference) of radicals such as 7root(2) + 3root(2), we proceed just as when combining like terms. Thus,

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

and 

7root(2) + 3root(2) = (7 + 3)root(2) = 10root(2)

Similarly,

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

To add root(75) + root(27), we simplify each radical first, then combine like radicals if possible.

root(75)+ root(27)=root(25·3)+root(9·3)=5root(3)) + 3root(3)=8root(3)

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

root(16x) + root(4x)= root(16)·root(x) +root(4)· root(x) = 4root(x) + 2root(x) = 6root(x)

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

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

root(2)[[root(7)+root(3)=root(2)root(7)+root(2)root(3)=root(14)+root(6)

And, with binomials,

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

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

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

Examples

Find the following products and simplify.

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

2. (root(2) + 4)(root(2) - 4) = (root(2))^2 - 4^2 = 2 - 16 = -14

3. (root(5)+root(3))(root(5)+root(3))=root(5)root(5) + 2root(5)root(3) + root(3)root(3)=5+2root(15)+3=8+2root(15)

Rationalizing Denominators

Each of the expressions, 5/root(3), root(7)/root(8), and 2/(3-root(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/root(3) = (5·root(3))/(root(3)·root(3)) = (5root(3))/3

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

4/root(x) = (4·root(x))/(root(x)·root(x)) = (4root(x))/x

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

root(7)/root(8) = (root(7)·root(2))/(root(8)·root(2))= root(14)/4

Multiply the numerator and denominator by root(2) because [root(8)·root(2) = root(16) and 16 is a perfect square number.

If we had multiplied by root(8), the results would have been the same, but the fraction would have to be reduced.

root(7)/root(8) = root(7)·root(8))/(root(8)·root(8)) = root(56)/8 = (root(4)·root(14))/8=(2root(14))/8 = root(14)/4

Before trying to rationalize the denominator for 2/(3-root(2)), recall that the product (a + b)(a - b) results in the difference of two squares:

(a + b)(a - b) = a^2 - b^2

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.

2/(3-root(2)) = (2(3+root(2)))/((3-root(2))(3+root(2))) 3 +root(2) is the conjugate surd of 3 - root(2).

                         = (2(3+root(2)))/(3^2-(root(2))^2)

The denominator is the difference of two squares.

                         = (2(3+root(2)))/(9-2)

The denominator is a rational number.

                         = (2(3+root(2)))/7