Multiply Square Roots

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You can multiply square roots, a type of radical expression, just as you might multiply whole numbers. Sometimes square roots have coefficients (an integer in front of the radical sign), but this only adds a step to the multiplication and does not change the process. The trickiest part of multiplying square roots is simplifying the expression to reach your final answer, but even this step is easy if you know your perfect squares.

Steps

Multiplying Square Roots Without Coefficients

  1. Multiply the radicands. A radicand is a number underneath the radical sign.[1] To multiply radicands, multiply the numbers as if they were whole numbers. Make sure to keep the product under one radical sign.[2]
    • For example, if you are calculating <math>\sqrt{15} \times \sqrt{5}</math>, you would calculate <math>15 \times 5 = 75</math>. So, <math>\sqrt{15} \times \sqrt{5} = \sqrt{75}</math>.
  2. Factor out any perfect squares in the radicand. To do this, see whether any perfect square is a factor of the radicand.[3] If you cannot factor out a perfect square, your answer is already simplified and you need not do anything further.
    • A perfect square is the result of multiplying an integer (a positive or negative whole number) by itself.[4] For example, 25 is a perfect square, because <math>5 \times 5 = 25</math>.
    • For example, <math>\sqrt{75}</math> can be factored to pull out the perfect square 25:
      <math>\sqrt{75}</math>
      =<math>\sqrt{25 \times 3}</math>
  3. Place the square root of the perfect square in front of the radical sign. Keep the other factor under the radical sign. This will give you your simplified expression.
    • For example, <math>\sqrt{75}</math> can be factored as <math>\sqrt{25 \times 3}</math>, so you would pull out the square root of 25 (which is 5):
      <math>\sqrt{75}</math>
      = <math>\sqrt{25 \times 3}</math>
      = <math>5\sqrt{3}</math>
  4. Square a square root. In some instances, you will need to multiply a square root by itself. Squaring a number and taking the square root of a number are opposite operations; thus, they undo each other. The result of squaring a square root, then, is simply the number under the radical sign.[5]
    • For example, <math>\sqrt{25} \times \sqrt{25} = 25</math>. You get that result because <math>\sqrt{25} \times \sqrt{25} = 5 \times 5 = 25</math>.

Multiplying Square Roots With Coefficients

  1. Multiply the coefficients. A coefficient is a number in front of the radical sign. To do this, just ignore the radical sign and radicand, and multiply the two whole numbers. Place their product in front of the first radical sign.
    • Pay attention to positive and negative signs when multiplying coefficients. Don't forget that a negative times a positive is a negative, and a negative times a negative is a positive.
    • For example, if you are calculating <math>3\sqrt{2} \times 2\sqrt{6}</math>, you would first calculate <math>3 \times 2 = 6</math>. So now your problem is <math>6\sqrt{2} \times \sqrt{6}</math>.
  2. Multiply the radicands. To do this, multiply the numbers as if they were whole numbers. Make sure to keep the product under the radical sign.
    • For example, if the problem is now <math>6\sqrt{2} \times \sqrt{6}</math>, to find the product of the radicands, you would calculate <math>2 \times 6 = 12</math>, so <math>\sqrt{2} \times \sqrt{6} = \sqrt{12}</math>. The problem now becomes <math>6\sqrt{12}</math>.
  3. Factor out any perfect squares in the radicand, if possible. You need to do this to simplify your answer.[6] If you cannot pull out a perfect square, your answer is already simplified and you can skip this step.
    • A perfect square is the result of multiplying an integer (a positive or negative whole number) by itself.[7] For example, 4 is a perfect square, because <math>2 \times 2 = 4</math>.
    • For example, <math>\sqrt{12}</math> can be factored to pull out the perfect square 4:
      <math>\sqrt{12}</math>
      =<math>\sqrt{4 \times 3}</math>
  4. Multiply the square root of the perfect square by the coefficient. Keep the other factor under the radicand. This will give you your simplified expression.
    • For example, <math>6\sqrt{12}</math> can be factored as <math>6\sqrt{4 \times 3}</math>, so you would pull out the square root of 4 (which is 2) and multiply it by 6:
      <math>6\sqrt{12}</math>
      = <math>6\sqrt{4 \times 3}</math>
      = <math>6 \times 2\sqrt{3}</math>
      = <math>12\sqrt{3}</math>

Tips

  • Always remember your perfect squares because it will make the process much easier!
  • Follow the usual sign rules to determine whether the new coefficient should be positive or negative. A positive coefficient multiplied by a negative coefficient will be negative. Two positive coefficients multiplied together or two negative coefficients multiplied together will be positive.
  • All terms under the radicand are always positive, so you will not have to worry about sign rules when multiplying radicands.

Things You'll Need

  • Pencil
  • Paper
  • Calculator

Related Articles

Sources and Citations

  1. ↑ http://www.mathwords.com/r/radicand.htm
  2. ↑ http://www.virtualnerd.com/pre-algebra/real-numbers-right-triangles/squares-square-roots/square-root-examples/multiplication-example
  3. ↑ http://www.uis.edu/ctl/wp-content/uploads/sites/76/2013/03/Radicals.pdf
  4. ↑ http://www.mathwarehouse.com/arithmetic/numbers/what-is-a-perfect-square.php
  5. ↑ http://www.virtualnerd.com/algebra-1/algebra-foundations/squaring-square-roots.php
  6. ↑ http://www.uis.edu/ctl/wp-content/uploads/sites/76/2013/03/Radicals.pdf
  7. ↑ http://www.mathwarehouse.com/arithmetic/numbers/what-is-a-perfect-square.php
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