DIFFERENCES OF TWO SQUARES(SURD)

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The product of the sum and difference of two squares of a surd is equal to the difference of the squares.

This means
$(a+\sqrt{b})(a-\sqrt{b})=(a)^2-(\sqrt{b})^2$

With the knowledge of this, let's try some examples

Example 1
Evaluate $(4\sqrt{11}+6)(4\sqrt{11}-6)$

Solution:
This is a difference of two squares
$(4\sqrt{11})^2-(6)^2$
$(4)^2(\sqrt{11})^2-(6)^2$
$16\times11-36$
$176-36=140$

Example 2
Evaluate $(2\sqrt{2}-4)(4+2\sqrt{3})$

Solution:
Let's rearrange this.
$(2\sqrt{3}-4)(2\sqrt{3}+4)$

This is a difference of two squares
$(2\sqrt{3})^2-(4)^2$
$(2)^2(\sqrt{3})^2-16$
$4\times3-16$
$12-16=-4$

Example 3
Evaluate $(3\sqrt{5}+4)(3\sqrt{5}-4)$

Solution:
This is a difference of two squares
$(3\sqrt{5})^2-(4)^2$
$(3)^2(\sqrt{5})^2-(16$
$9\times5-16$
$45-16=29$

Example 4
Evaluate $(2+\sqrt{3})(4-\sqrt{12})$

Solution
$\sqrt{12}$ can be simplify further, hence
$(2+\sqrt{3})(4-\sqrt{4\times3})$
$(2+\sqrt{3})(4-\sqrt{4}\times\sqrt{3})$
$(2+\sqrt{3})(4-2\sqrt{3})$
$(2+\sqrt{3})-(2(2-\sqrt{3}))$

If $(2(2-\sqrt{3}))$ can be rewritten as $(2-\sqrt{3})(2)$, hence
$(2+\sqrt{3})-(2-\sqrt{3})(2)$

We have difference of two squares
$(2^2-\sqrt{3}^2)(2)$
$(4-3)(2)$
$(1)(2)$
$2$

Example 5
Evaluate $(3\sqrt{5}+2)(3\sqrt{5}-2)$

Solution
This is a difference of two squares
$(3\sqrt{5})^2-(2)^2$
$9(\sqrt{5})^2-4$
$9\times5-4$
$45-4=41$

Example 6
Evaluate $(3\sqrt{3}-3)(3\sqrt{3}+3)$

Solution
This is a difference of two squares
$(3\sqrt{3})^2-(3)^2$
$9(\sqrt{3})^2-(3)^2$
$9\times3-9$
$27-9=18$

Example 7
Simplify $(\sqrt{5+2\sqrt{6}})((\sqrt{5-2\sqrt{6}})$

Solution
$(\sqrt{5+2\sqrt{6}})((\sqrt{5-2\sqrt{6}})$ can be rewritten as:
$\sqrt{5+2\sqrt{6}\times5-2\sqrt{6}}$

$\sqrt{25-4(\sqrt{6})2}$
$\sqrt{25-4(6)}$
$\sqrt{25-24}$
$\sqrt{1}$
$1$


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