Challenge Level

For each of the statements provided below, determine which non-negative values of $a$, $b$, $c$, and $d$, if any, make the equation true.

These can be attempted in any order but you might find that some statements can help inform your decisions about others.

You can download these statements as a set of cards that can be cut out and considered in any order.

a) $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$

b) $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$

c) $\sqrt{23-6\sqrt{6-4\sqrt{2}}}=\sqrt{a}+\sqrt{b}$

d) $a\sqrt{b}=\sqrt{ab}$

e) $\dfrac{\sqrt{ab}}{\sqrt{a}+\sqrt{b}}=1$

f) $\sqrt{a} - \sqrt{b}=\sqrt{a - b}$

g) $\sqrt{a}+\sqrt{b}=\sqrt{a+b+\sqrt{4ab}}$

h) $\dfrac{\sqrt{a}+b}{\sqrt{c}+d}=(\sqrt{a}+b)(\sqrt{c}-d)$

i) $\sqrt{5+2\sqrt{6}}=\sqrt{a}+\sqrt{b}$