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:

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.

\((3 + \sqrt{2})(2 + \sqrt{5}) = 6 + 2\sqrt{2} + 3\sqrt{5} + \sqrt{10}\)

The same method can be used if the numbers in the surds are the same:

Simplify fully \((1 + \sqrt{3})(5 - \sqrt{3})\)

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

\((7 + \sqrt{3})(8 + \sqrt{2})\)
\((4 - \sqrt{6})(3 + \sqrt{6})\)

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Surds - Higher - OCRMultiplying out brackets including surds