For example, if you are calculating 15×5{\displaystyle {\sqrt {15}}\times {\sqrt {5}}}, you would calculate 15×5=75{\displaystyle 15\times 5=75}. So, 15×5=75{\displaystyle {\sqrt {15}}\times {\sqrt {5}}={\sqrt {75}}}.
A perfect square is the result of multiplying an integer (a positive or negative whole number) by itself. [4] X Research source For example, 25 is a perfect square, because 5×5=25{\displaystyle 5\times 5=25}. For example, 75{\displaystyle {\sqrt {75}}} can be factored to pull out the perfect square 25:75{\displaystyle {\sqrt {75}}}=25×3{\displaystyle {\sqrt {25\times 3}}}
For example, 75{\displaystyle {\sqrt {75}}} can be factored as 25×3{\displaystyle {\sqrt {25\times 3}}}, so you would pull out the square root of 25 (which is 5):75{\displaystyle {\sqrt {75}}}= 25×3{\displaystyle {\sqrt {25\times 3}}}= 53{\displaystyle 5{\sqrt {3}}}
For example, 25×25=25{\displaystyle {\sqrt {25}}\times {\sqrt {25}}=25}. You get that result because 25×25=5×5=25{\displaystyle {\sqrt {25}}\times {\sqrt {25}}=5\times 5=25}.
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 32×26{\displaystyle 3{\sqrt {2}}\times 2{\sqrt {6}}}, you would first calculate 3×2=6{\displaystyle 3\times 2=6}. So now your problem is 62×6{\displaystyle 6{\sqrt {2}}\times {\sqrt {6}}}.
For example, if the problem is now 62×6{\displaystyle 6{\sqrt {2}}\times {\sqrt {6}}}, to find the product of the radicands, you would calculate 2×6=12{\displaystyle 2\times 6=12}, so 2×6=12{\displaystyle {\sqrt {2}}\times {\sqrt {6}}={\sqrt {12}}}. The problem now becomes 612{\displaystyle 6{\sqrt {12}}}.
A perfect square is the result of multiplying an integer (a positive or negative whole number) by itself. [9] X Research source For example, 4 is a perfect square, because 2×2=4{\displaystyle 2\times 2=4}. For example, 12{\displaystyle {\sqrt {12}}} can be factored to pull out the perfect square 4:12{\displaystyle {\sqrt {12}}}=4×3{\displaystyle {\sqrt {4\times 3}}}
For example, 612{\displaystyle 6{\sqrt {12}}} can be factored as 64×3{\displaystyle 6{\sqrt {4\times 3}}}, so you would pull out the square root of 4 (which is 2) and multiply it by 6:612{\displaystyle 6{\sqrt {12}}}= 64×3{\displaystyle 6{\sqrt {4\times 3}}}= 6×23{\displaystyle 6\times 2{\sqrt {3}}}= 123{\displaystyle 12{\sqrt {3}}}
