Multiplying out brackets including surds - Higher

Expressions with brackets that include surds can be multiplied out or expanded.

Multiply out (2 + \sqrt{5}) (3 + \sqrt{2})

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:

A grid that has simplified (3 + √2)(2 + √5)

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.

(2 + \sqrt{5})(3 + \sqrt{2}) = 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 (5 - \sqrt{3})(1 + \sqrt{3})

Surd table showing 5 minus root 3 add 1 plus root 3

(5 – \sqrt{3}) (1 + \sqrt{3})= 5 – \sqrt{3}  + 5\sqrt{3} – 3

= 4{\sqrt{3}} + 2

Question
  1. Expand (7 + \sqrt{3})(8 + \sqrt{2})
  2. Expand and simplify (4 - \sqrt{6})(3 + \sqrt{6})
  1. 56 + 8\sqrt{3} + 7\sqrt{2} + \sqrt{6}
  2. 12 + 4\sqrt{6} - 3\sqrt{6} – 6 = 6 + \sqrt{6}