Multiplying out brackets including surds
Expressions with brackets that include surds, for example \(\sqrt{11}(2-\sqrt{3})\), can be multiplied out in a similar way to multiplying out algebraic expressions, \(\sqrt{11}(2-\sqrt{3}) = 2\sqrt{11}-\sqrt{33}\).
Example
Simplify fully \((3 + \sqrt{2})(2 + \sqrt{5})\)
Each term in the first bracket has to be multiplied by each term in the second bracket. One way to do this is to use a grid:
The four terms cannot be simplified because each of the surds has a different number inside the square root, and none of the surds can be simplified.
\((3 + \sqrt{2})(2 + \sqrt{5}) = 6 + 2\sqrt{2} + 3\sqrt{5} + \sqrt{10}\)
The same method can be used if the numbers in the surds are the same:
Simplify fully \((1 + \sqrt{3})(5 - \sqrt{3})\)
The surds have the same number inside the square root, so they give a rational number when multiplied together. The four terms can be simplified by adding together the rational terms and the irrational terms:
\((5) + (-3) = 2 \) and \((5\sqrt{3}) + (-\sqrt{3}) = 4\sqrt{3}\), so \((1 + \sqrt{3})(5 - \sqrt{3}) = 2 + 4\sqrt{3}\)
Question
Simplify fully the following:
- \((7 + \sqrt{3})(8 + \sqrt{2})\)
- \((4 - \sqrt{6})(3 + \sqrt{6})\)
- \(56 + 8\sqrt{3} + 7\sqrt{2} + \sqrt{6}\)
- \(12 + 4\sqrt{6} - 3\sqrt{6} 鈥 6 = 6 + \sqrt{6}\)