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Surds: Multiplying Brackets

Sometimes, we need to multiply brackets together which contain surds.

Some surd expressions contain brackets.

To simplify them, we expand our brackets.

Let's look at an example:

$5(3 + \sqrt{8})$

Multiply each term inside the bracket by $5$ .

Inside the brackets we have two terms: $3$ and $\sqrt{8}$ .

Multiply $5 \times 3$

Multiply $5 \times \sqrt{8}$

Now rewrite the expression without brackets.

$15 + 5\sqrt{8}$ or we could also write $5\sqrt{8} + 15$ .

Expand $7(\sqrt{5} + 2)$

Sometimes, there might be two sets of brackets.

To expand two brackets, remember the acronym: FOIL: First, Outside, Inside, Last.

Expand and simplify: $(5+\sqrt{2})(3+4\sqrt{2})$

Multiply the first terms together:

$({\color{#21affb}5} + \sqrt{2})({\color{#21affb}3} + 4\sqrt{2})$ gives us $5 \times 3 = 15$

Multiply the outside terms: $({\color{#21affb}5} + \sqrt{2})(3 + {\color{#21affb}4\sqrt{2}})$

Multiply the inside terms: $(5 + {\color{#21affb}\sqrt{2}})({\color{#21affb}3} + 4\sqrt{2})$

Multiply the last terms: $(5 + {\color{#21affb}\sqrt{2}})(3 + {\color{#21affb}4\sqrt{2}})$

$\sqrt{2} \times 4\sqrt{2} = 4 \times \sqrt{4} = 4 \times 2 = 8$

Put all four terms together

$15 + 20\sqrt{2} + 3\sqrt{2} + 8$

Now simplify the like-terms.

Our final answer is $23 + 23\sqrt{2}$ or $23\sqrt{2} + 23$

What is $(2+\sqrt{2})(\sqrt{2}+4)$ ?

Let's try a harder example:

Expand $(4\sqrt{2} + \sqrt{3})(3\sqrt{5} - 2\sqrt{3})$

Multiply the first terms: $4\sqrt{2} \times 3\sqrt{5}$

Multiply the outside terms

Multiply the inside terms

$\sqrt{3} \times 3\sqrt{5} = 3\sqrt{15}$

Multiply the last terms: $\sqrt{3} \times -2\sqrt{3}$

Put all four terms together

$12\sqrt{10} - 8\sqrt{6} + 3\sqrt{15} - 6$

As there are no like terms, this can not be simplified

So $12\sqrt{10} - 8\sqrt{6} + 3\sqrt{15} - 6$ is the final answer!