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Adding (and Subtracting) Square Roots (page 3 of 7)

Sections: Square roots, More simplification / Multiplication, Adding (and subtracting) square roots, Conjugates / Dividing by square roots, Rationalizing denominators, Higher-Index Roots, A special case of rationalizing / Radicals & exponents / Radicals & domains


Adding (and Subtracting) Square Roots

Just as with "regular" numbers, square roots can be added together. But you might not be able to simplify the addition all the way down to one number. Just as "you can't add apples and oranges", so also you cannot combine "unlike" radicals. To add radical terms together, they have to have the same radical part.

  • Simplify:  2sqrt(3) + 3sqrt(3)
  • Since the radical is the same in each term (namely, the square root of three), I can combine the terms. I have two copies of the radical, added to another three copies. This gives me five copies:

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

That middle step, with the parentheses, shows the reasoning that justifies the final answer. You probably won't ever need to "show" this step, but it's what should be going through your mind.

  • Simplify:  sqrt(3) + 4sqrt(3)
  • The radical part is the same in each term, so I can do this addition. To help me keep track that the first term means "one copy of the square root of three", I'll insert the "understood" "1":

     

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      sqrt(3) + 4 sqrt(3) = 1 sqrt(3) + 4 sqrt(3) = (1 + 4) sqrt(3) = 5 sqrt(3)

Don't assume that expressions with unlike radicals cannot be simplified. It is possible that, after simplifying the radicals, the expression can indeed be simplified.

  • Simplify:  sqrt(9) + sqrt(25)
  • To simplify a radical addition, I must first see if I can simplify each radical term. In this particular case, the square roots simplify "completely" (that is, down to whole numbers):

      sqrt(9) + sqrt(25) = 3 + 5 = 8

  • Simplify:  3 sqrt[4] + 2 sqrt[4]

    I have three copies of the radical, plus another two copies, giving me— Wait a minute! I can simplify those radicals right down to whole numbers:

      3 sqrt[4] + 2 sqrt[4] = 3 * 2 + 2 * 2 = 6 + 4 = 10

Don't worry if you don't see a simplification right away. If I hadn't noticed until the end that the radical simplified, my steps would have been different, but my final answer would have been the same:

    3 sqrt[4] + 2 sqrt[4] = 5 sqrt[4] = 5 * 2 = 10

  • Simplify:  3 sqrt[3] + 2 sqrt[5] + sqrt[3]
  • I can only combine the "like" radicals, so I'll end up with two terms in my answer:

      3 sqrt[3] + 2 sqrt[5] + sqrt[3] = 3 sqrt[3] + 1 sqrt[3] + 2 sqrt[5] = 4 sqrt[3] + 2 sqrt[5]

There is not, to my knowledge, any preferred ordering of terms in this sort of expression, so the expression 2 sqrt[5] + 4 sqrt[3] should also be an acceptable answer.

  • Simplify:  3 sqrt[8] + 5 sqrt[2]   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved
  • I can simplify the radical in the first term, and this will create "like" terms:

      3 sqrt[8] + 5 sqrt[2] = 3 sqrt[(2 * 2) * 2] + 5 sqrt[2] = 3 * 2 sqrt[2] + 5 sqrt[2] = 6 sqrt[2] + 5 sqrt[2] = 11 sqrt[2]

  • Simplify:  sqrt[18] - 2 sqrt[27] + 3 sqrt[3] - 6 sqrt[8]
  • I can simplify most of the radicals, and this will allow for at least a little simplification:

      sqrt[18] - 2 sqrt[27] + 3 sqrt[3] - 6 sqrt[8] = sqrt[(3 * 3) * 2] - 2 sqrt[(3 * 3) *3)] + 3 sqrt[3] - 6 sqrt[(2 * 2) * 2] = 3 sqrt[2] - 6 sqrt[3] + 3 sqrt[3] - 12 sqrt[2] = -9 sqrt[2] - 3 sqrt[3]

  • Simplify:  2 sqrt(3) + 3 sqrt(5)
  • These two terms have "unlike" radical parts, and I can't take anything out of either radical. Then I can't simplify the expression 2sqrt(3) + 3sqrt(5) any further and my answer has to be:

      2 sqrt(3) + 3 sqrt(5)
      (expression is already fully simplified)

  • Expand:  sqrt[2](3 + sqrt[3])
  • To expand (that is, to multiply out and simplify) this expression, I first need to take the square root of two through the parentheses:

      sqrt[2](3 + sqrt[3]) = 3 sqrt[2] + sqrt[2 * 3] = 3 sqrt[2] + sqrt[6]

As you can see, the simplification involved turning a product of radicals into one radical containing the value of the product (being 2×3 = 6). You should expect to need to manipulate radical products in both "directions".

  • Expand:  sqrt(3) (2sqrt(3) + sqrt(5))
    • sqrt(3) (2 sqrt(3) + sqrt(5)) = sqrt(3) * 2 * sqrt(3) + sqrt(3) sqrt(5) = 2 sqrt(3 * 3) + sqrt(3 * 5) = 2 * 3 + sqrt(15) = 6 + sqrt(15)

  • Expand:  (1 + sqrt[2])(3 - sqrt[2])
  • It will probably be simpler to do this multiplication "vertically".

      (1 + sqrt[2])(3 - sqrt[2]) = -1sqrt[2] - sqrt[2]sqrt[2] + 3 + 3sqrt[2] = 3 + 2sqrt[2] - sqrt[2 * 2]

    Simplifying gives me:  3 + 2sqrt[2] - 2 = 1 + 2 sqrt[2]

By doing the multiplication vertically, I could better keep track of my steps. You should use whatever multiplication method works best for you.

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Cite this article as:

Stapel, Elizabeth. "Adding (and Subtracting) Square Roots." Purplemath. Available from
    http://www.purplemath.com/modules/radicals3.htm. Accessed
 

 

 

 

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