Question

Match the following to their simplified values.

• A. 11 + 6$\sqrt{2}$
• B. $2+\sqrt{3}$
• C. 4

Answer 1

Using the identity, (a+b)(a-b) = a${}^2$-b${}^2$,
$(3-\sqrt{5})\times (3+\sqrt{5})$
=$3^2-(\sqrt{5}{)}^2$
= $9-5$
= $4$

In order to rationalise $\frac{1}{2-\sqrt{3}}$,
we multiply the numerator and denominator by $(2+\sqrt{3})$ .

$\frac{1}{2-\sqrt{3}}=\frac{1}{2-\sqrt{3}}\times \frac{2+\sqrt{3}}{2+\sqrt{3}}$

$=\frac{2+\sqrt{3}}{2^2-{\sqrt{3}}^2}=\frac{2+\sqrt{3}}{1}=2+\sqrt{3}$.

Using the identity,
(a+b)${}^2$ = a${}^2$+ 2ab + b${}^2$,

$(3+\sqrt{2}{)}^2=3^2+{\sqrt{2}}^2+2(3)\sqrt{2}$
$=9+2+6\sqrt{2}$
$=11+6\sqrt{2}$.