1. ## Surds (Q3)

Without using a calculator, find the values of the integers $\displaystyle a$ and $\displaystyle b$ for which the solution of the equation

$\displaystyle x\sqrt{24} = x\sqrt{3} + \sqrt{6}$

is $\displaystyle \frac{a+\sqrt{b}}{7}$

2. Originally Posted by Punch
Without using a calculator, find the values of the integers $\displaystyle a$ and $\displaystyle b$ for which the solution of the equation

$\displaystyle x\sqrt{24} = x\sqrt{3} + \sqrt{6}$

is $\displaystyle \frac{a+\sqrt{b}}{7}$
$\displaystyle x=\frac{\sqrt{6}}{\sqrt{24}\sqrt{3}}$

Rationalize the surds and you will get

$\displaystyle \frac{\sqrt{144}+\sqrt{18}}{21}$

$\displaystyle \frac{12+3\sqrt{2}}{21}$
...

3. Hello, Punch!

$\displaystyle \text{Without using a calculator, find the values of }a\text{ and }b$

$\displaystyle \text{for which the solution of: }\:x\sqrt{24} \:=\: x\sqrt{3} + \sqrt{6}$

. . $\displaystyle \text{is: }\;x\:=\:\dfrac{a+\sqrt{b}}{7}$

We have: .$\displaystyle x\sqrt{24} \;=\;x\sqrt{3} + \sqrt{6} \quad\Rightarrow\quad x\sqrt{24} - x\sqrt{3} \;=\;\sqrt{6}$

. . . . . $\displaystyle (\sqrt{24}-\sqrt{3})x \;=\;\sqrt{6} \quad\Rightarrow\quad x \;=\;\dfrac{6}{\sqrt{24}-\sqrt{3}}$

Factor and reduce:

. . $\displaystyle x \;=\;\dfrac{\sqrt{2}\!\cdot\!\sqrt{3}}{2\!\cdot\!\ sqrt{2}\!\cdot\!\sqrt{3} - \sqrt{3}} \;=\;\dfrac{\sqrt{3}\!\cdot\!\sqrt{2}} {\sqrt{3}\!\cdot\!(2\sqrt{2}-1)} \;=\; \dfrac{\sqrt{2}}{2\sqrt{2}-1}$

Rationalize:

. . $\displaystyle x \;=\;\dfrac{\sqrt{2}}{2\sqrt{2}-1}\cdot\dfrac{2\sqrt{2}+1}{2\sqrt{2}+1} \;=\;\dfrac{4 + \sqrt{2}}{8-1} \;=\;\dfrac{4 + \sqrt{2}}{7}$

Therefore: .$\displaystyle a = 4,\;\;\rlap{/////}b = 1 \;\;b = 2$

Edit: Right, masters! . . . b = 2 . . . Boy, that was a dumb error!

4. $\displaystyle b=2$