Multiply Square Roots
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.
Contents
Steps
Multiplying Square Roots Without Coefficients
- Multiply the radicands. A radicand is a number underneath the radical sign.
- 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>.
To multiply radicands, multiply the numbers as if they were whole numbers. Make sure to keep the product under one radical sign.
- Factor out any perfect squares in the radicand. To do this, see whether any perfect square is a factor of the radicand.
- A perfect square is the result of multiplying an integer (a positive or negative whole number) by itself. 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>
If you cannot factor out a perfect square, your answer is already simplified and you need not do anything further.
- 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>
- 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):
- 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.
- 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
- 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>.
- 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>.
- Factor out any perfect squares in the radicand, if possible. You need to do this to simplify your answer.
- A perfect square is the result of multiplying an integer (a positive or negative whole number) by itself. 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>
If you cannot pull out a perfect square, your answer is already simplified and you can skip this step.
- 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>
- 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:
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
- Multiply Perennials by Dividing Them
- Multiply Radicals
- Multiply Binomials Using the FOIL Method
- Simplify a Square Root
- Add and Subtract Square Roots
- Divide Square Roots
- Calculate a Square Root by Hand
Sources and Citations
- http://www.mathwords.com/r/radicand.htm
- http://www.virtualnerd.com/pre-algebra/real-numbers-right-triangles/squares-square-roots/square-root-examples/multiplication-example
- ↑ http://www.uis.edu/ctl/wp-content/uploads/sites/76/2013/03/Radicals.pdf
- ↑ http://www.mathwarehouse.com/arithmetic/numbers/what-is-a-perfect-square.php
- http://www.virtualnerd.com/algebra-1/algebra-foundations/squaring-square-roots.php
What links here
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