Question

Find the value of a in the following: 6 /
3√2 -2√3 = 3√2-a√3​

Answer 1

Answer:

$\dfrac{6}{3\sqrt{2}-2\sqrt{3}}=3\sqrt{2}-a\sqrt{3}\quad :\quad a=-2$

Step-by-step explanation:

$\dfrac{6}{3\sqrt{2}-2\sqrt{3}}=3\sqrt{2}-a\sqrt{3}$

$\mathrm{Switch\:sides}$

$3\sqrt{2}-a\sqrt{3}=\dfrac{6}{3\sqrt{2}-2\sqrt{3}}$

$\mathrm{Simplify\:}\dfrac{6}{3\sqrt{2}-2\sqrt{3}}$

$\dfrac{6}{3\sqrt{2}-2\sqrt{3}}$

$\mathrm{Factor}\:3\sqrt{2}-2\sqrt{3}$

$3\sqrt{2}-2\sqrt{3}$

$3=\sqrt{3}\sqrt{3}$

$=\sqrt{3}\sqrt{3}\sqrt{2}-2\sqrt{3}$

$\mathrm{Factor\:out\:common\:term\:}\sqrt{3}$

$=\sqrt{3}\left(\sqrt{3}\sqrt{2}-2\right)$

$\mathrm{Refine}$

$=\sqrt{3}\left(\sqrt{6}-2\right)$

$=\dfrac{6}{\sqrt{3}\left(\sqrt{6}-2\right)}$

$\mathrm{Factor}\:6:\quad 2\cdot \:3$

$=\dfrac{2\cdot \:3}{\sqrt{3}\left(\sqrt{6}-2\right)}$

$\mathrm{Cancel\:}\dfrac{2\cdot \:3}{\sqrt{3}\left(\sqrt{6}-2\right)}:\quad \dfrac{2\sqrt{3}}{\sqrt{6}-2}$

$=\dfrac{2\sqrt{3}}{\sqrt{6}-2}$

$3\sqrt{2}-a\sqrt{3}=\dfrac{2\sqrt{3}}{\sqrt{6}-2}$

$\mathrm{Subtract\:}3\sqrt{2}\mathrm{\:from\:both\:sides}$

$3\sqrt{2}-a\sqrt{3}-3\sqrt{2}=\dfrac{2\sqrt{3}}{\sqrt{6}-2}-3\sqrt{2}$

$\mathrm{Simplify}$

$3\sqrt{2}-a\sqrt{3}-3\sqrt{2}=\dfrac{2\sqrt{3}}{\sqrt{6}-2}-3\sqrt{2}$

$\mathrm{Simplify\:}3\sqrt{2}-a\sqrt{3}-3\sqrt{2}:\quad -a\sqrt{3}$

$\mathrm{Simplify\:}\dfrac{2\sqrt{3}}{\sqrt{6}-2}-3\sqrt{2}:\quad 3\sqrt{2}+2\sqrt{3}-3\sqrt{2}$

$-a\sqrt{3}=3\sqrt{2}+2\sqrt{3}-3\sqrt{2}$

$\mathrm{Divide\:both\:sides\:by\:}-\sqrt{3}$

$\dfrac{-a\sqrt{3}}{-\sqrt{3}}=\dfrac{3\sqrt{2}}{-\sqrt{3}}+\dfrac{2\sqrt{3}}{-\sqrt{3}}-\dfrac{3\sqrt{2}}{-\sqrt{3}}$

$\mathrm{Simplify}$

$\mathrm{Simplify\:}\dfrac{-a\sqrt{3}}{-\sqrt{3}}:\quad a$

$\mathrm{Simplify\:}\dfrac{3\sqrt{2}}{-\sqrt{3}}+\dfrac{2\sqrt{3}}{-\sqrt{3}}-\dfrac{3\sqrt{2}}{-\sqrt{3}}$

$\mathrm{Group\:like\:terms}$

$=\dfrac{3\sqrt{2}}{-\sqrt{3}}-\dfrac{3\sqrt{2}}{-\sqrt{3}}+\dfrac{2\sqrt{3}}{-\sqrt{3}}$

$\mathrm{Apply\:rule}\:\dfrac{a}{c}\pm \dfrac{b}{c}=\dfrac{a\pm \:b}{c}$

$=\dfrac{3\sqrt{2}-3\sqrt{2}+2\sqrt{3}}{-\sqrt{3}}$

$\mathrm{Add\:similar\:elements:}\:3\sqrt{2}-3\sqrt{2}=0$

$=\dfrac{2\sqrt{3}}{-\sqrt{3}}$

$\mathrm{Apply\:the\:fraction\:rule}:\quad \dfrac{a}{-b}=-\dfrac{a}{b}$

$=-\dfrac{2\sqrt{3}}{\sqrt{3}}$

$\mathrm{Cancel\:the\:common\:factor:}\:\sqrt{3}$

$=-2$

$a=-2$