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 The square root of a number is a number which, when multiplied by itself, gives the original number. So the square root of 25 is 5 (5 脳 5 = 25)., 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 A number that can be written in fraction form. This includes integers, terminating decimals, repeating decimals and fractions. 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:
- \((7 + \sqrt{3})(8 + \sqrt{2})\)
- \((4 - \sqrt{6})(3 + \sqrt{6})\)
- \(56 + 8\sqrt{3} + 7\sqrt{2} + \sqrt{6}\)
- \(12 + 4\sqrt{6} - 3\sqrt{6} - 6 = 6 + \sqrt{6}\)