Difference between revisions of "Multiply Square Roots"
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== 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. | ||
