Difference between revisions of "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. | 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. | ||
| − | [[Category:Multiplication and Division]] | + | [[Category: Multiplication and Division]] |
== Steps == | == Steps == | ||
===Multiplying Square Roots Without Coefficients=== | ===Multiplying Square Roots Without Coefficients=== | ||
| − | #Multiply the radicands. A radicand is a number underneath the radical sign.<ref>http://www.mathwords.com/r/radicand.htm</ref> To multiply radicands, multiply the numbers as if they were whole numbers. Make sure to keep the product under one radical sign.<ref>http://www.virtualnerd.com/pre-algebra/real-numbers-right-triangles/squares-square-roots/square-root-examples/multiplication-example</ref> | + | #Multiply the radicands. A radicand is a number underneath the radical sign.<ref name="rf16525">http://www.mathwords.com/r/radicand.htm</ref> To multiply radicands, multiply the numbers as if they were whole numbers. Make sure to keep the product under one radical sign.<ref name="rf16526">http://www.virtualnerd.com/pre-algebra/real-numbers-right-triangles/squares-square-roots/square-root-examples/multiplication-example</ref> |
#*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>. | #*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>. | ||
| − | #Factor out any perfect squares in the radicand. To do this, see whether any perfect square is a factor of the radicand.<ref>http://www.uis.edu/ctl/wp-content/uploads/sites/76/2013/03/Radicals.pdf</ref> If you cannot factor out a perfect square, your answer is already simplified and you need not do anything further. | + | #Factor out any perfect squares in the radicand. To do this, see whether any perfect square is a factor of the radicand.<ref name="rf16527">http://www.uis.edu/ctl/wp-content/uploads/sites/76/2013/03/Radicals.pdf</ref> 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.<ref>http://www.mathwarehouse.com/arithmetic/numbers/what-is-a-perfect-square.php</ref> For example, 25 is a perfect square, because <math>5 \times 5 = 25</math>. | + | #*A perfect square is the result of multiplying an integer (a positive or negative whole number) by itself.<ref name="rf16528">http://www.mathwarehouse.com/arithmetic/numbers/what-is-a-perfect-square.php</ref> 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:<br><math>\sqrt{75}</math><br>=<math>\sqrt{25 \times 3}</math> | #*For example, <math>\sqrt{75}</math> can be factored to pull out the perfect square 25:<br><math>\sqrt{75}</math><br>=<math>\sqrt{25 \times 3}</math> | ||
#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. | #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):<br><math>\sqrt{75}</math><br>= <math>\sqrt{25 \times 3}</math><br>= <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):<br><math>\sqrt{75}</math><br>= <math>\sqrt{25 \times 3}</math><br>= <math>5\sqrt{3}</math> | ||
| − | #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.<ref>http://www.virtualnerd.com/algebra-1/algebra-foundations/squaring-square-roots.php</ref> | + | #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.<ref name="rf16529">http://www.virtualnerd.com/algebra-1/algebra-foundations/squaring-square-roots.php</ref> |
#*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>. | #*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>. | ||
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#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. | #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>. | #*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.<ref | + | #Factor out any perfect squares in the radicand, if possible. You need to do this to simplify your answer.<ref name="rf16527" /> 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.<ref | + | #*A perfect square is the result of multiplying an integer (a positive or negative whole number) by itself.<ref name="rf16528" /> 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:<br><math>\sqrt{12}</math><br>=<math>\sqrt{4 \times 3}</math> | #*For example, <math>\sqrt{12}</math> can be factored to pull out the perfect square 4:<br><math>\sqrt{12}</math><br>=<math>\sqrt{4 \times 3}</math> | ||
#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. | #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. | ||
