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Simplifying Radical Expressions
     Containing Parentheses (page 3 of 4)

Sections: Square roots, Other roots / Domains, Further simplifying, Rationalizing denominators

You have seen the simple case of radical multiplication, but you will probably have simplification problems that are more complex. Work with radicals in a manner similar to how you worked with polynomials. For instance, if you had to multiply 3(x + 2), you would take the 3 through the parentheses to get 3x + 6. Similarly:

  • Simplify 3(2 + sqrt(5))

      • 3(2 + sqrt(5)) = 6 + 3sqrt(5)

  • Simplify sqrt(3) (2 + sqrt(5))

      • sqrt(3) (2 + sqrt(5)) = 2sqrt(3) + sqrt(15)

  • Simplify sqrt(3) (2sqrt(3) + sqrt(5))

      • sqrt(3) (2sqrt(3) + sqrt(5)) = 6 + sqrt(15)

  • Simplify (sqrt(3) + sqrt(5)) (sqrt(3) - sqrt(6))

    You can do this multiplication vertically, just as you did with polynomials. You can do the multiplication horizontally, too, but I find vertical to be easier for me:

      multiplication

    Then I complete the calculations by simplifying:

      (sqrt(3) + sqrt(5)) (sqrt(3) - sqrt(6)) = 3 + sqrt(15) - 3sqrt(2) - sqrt(30)

  • Simplify (sqrt(3) + sqrt(5)) (sqrt(3) - sqrt(5))

    I do the multiplication:

      multiplication

    Then I simplify:

      (sqrt(3) + sqrt(5)) (sqrt(3) - sqrt(5)) = 3 - 5 = -2

Note in the last example above how I ended up with all whole numbers. (Okay, technically they're integers, but the point is that they are not radicals.) I had multiplied two radical "binomials" together and gotten an answer that contained no radicals. You may also have noticed that the two "binomials" were the same except for the sign in the middle. This is important.

    Given sqrt(a) + sqrt(b), the conjugate is sqrt(a) - sqrt(b).

That is, the conjugate has the same numbers, but has the opposite sign in the middle. In other words, not only is sqrt(a) - sqrt(b) the conjugate of sqrt(a) + sqrt(b), but sqrt(a) + sqrt(b) is the conjugate of sqrt(a) - sqrt(b). When you multiply conjugates, you are doing something similar to what happens with a difference of squares:

    a2 – b2 = (a + b)(a – b)   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

When you multiply the factors a + b and a – b, the middle "ab" terms cancel out:

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

The same thing happens when you multiply conjugates:

    (sqrt(a) + sqrt(b)) (sqrt(a) - sqrt(b)) = a - b

We will see shortly why this matters. To get to that point, let's take a look at fractions containing radicals in their denominators.

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Cite this article as:

Stapel, Elizabeth. "Simplifying Radical Expressions Containing Parentheses." Purplemath.
    Available from http://www.purplemath.com/modules/radicals3.htm.
    Accessed
 

 

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