Question

find the value of √[ a-b] if [(8+3sqrt(7))/(8-3sqrt(7)]-[(8-3sqrt(7))/(8+3sqrt(7)]=a-b√7​

Answer 1

Answer:

$\boxed{\red{\sf\:\sqrt{(\:a\:-\:b\:)}\:=\:4\:\sqrt{6}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle\sf\:a\:-\:b\:\sqrt{7}\:=\:\dfrac{8\:+\:3\:\sqrt{7}}{8\:-\:3\:\sqrt{7}}\:-\:\dfrac{8\:-\:3\:\sqrt{7}}{8\:+\:3\:\sqrt{7}}$

We have to find the value of $\displaystyle\sf\:\sqrt{a\:-\:b}$

$\displaystyle\sf\:a\:-\:b\:\sqrt{7}\:=\:\dfrac{8\:+\:3\:\sqrt{7}}{8\:-\:3\:\sqrt{7}}\:-\:\dfrac{8\:-\:3\:\sqrt{7}}{8\:+\:3\:\sqrt{7}}\\\\\\\implies\sf\:a\:-\:b\:\sqrt{7}\:=\:\dfrac{8\:+\:3\:\sqrt{7}\:\times\:8\:+\:3\:\sqrt{7}\:-\:8\:-\:3\:\sqrt{7}\:\times\:8\:-\:3\:\sqrt{7}}{8\:-\:3\:\sqrt{7}\:\times\:8\:+\:3\:\sqrt{7}}\:\:\:-\:-\:[\:Equating\:the\:denominator\:]\\\\\\\implies\sf\:a\:-\:b\:\sqrt{7}\:=\:\dfrac{\:(\:8\:+\:3\:\sqrt{7}\:)^2\:-\:(\:8\:-\:3\:\sqrt{7}\:)^2}{\:(\:8\:)^2\:-\:(\:3\:\sqrt{7}\:)^2}\:\:\:-\:-\:[\:(\:a\:+\:b\:)\:(\:a\:-\:b\:)\:=\:a^2\:-\:b^2\:]\\\\\\\implies\sf\:a\:-\:b\:\sqrt{7}\:=\:\dfrac{8^2\:+\:2\:\times\:8\:\times\:3\:\sqrt{7}\:+\:(\:3\:\sqrt{7}\:)^2\:-\:8^2\:-\:2\:\times\:8\:\times\:3\:\sqrt{7}\:+\:(\:3\:\sqrt{7}\:)^2}{64\:-\:9\:\times\:7}\:\:\:-\:-\:-\:[\:(\:a\:\pm\:b\:)^2\:=\:a^2\:\pm\:2ab\:+\:b^2\:]\\\\\\\implies\sf\:a\:-\:b\:\sqrt{7}\:=\:\dfrac{64\:+\:48\:\sqrt{7}\:+\:63\:-\:\left(\:64\:-\:48\:\sqrt{7}\:+\:63\:\right)}{64\:-\:63}\\\\\\\implies\sf\:a\:-\:b\:\sqrt{7}\:=\:\dfrac{\cancel{64}\:+\:48\:\sqrt{7}\:+\:\cancel{63}\:-\:\cancel{64}\:+\:48\:\sqrt{7}\:-\:\cancel{63}}{1}$

$\displaystyle\\\\\\\implies\sf\:a\:-\:b\:\sqrt{7}\:=\:48\:\sqrt{7}\:+\:48\:\sqrt{7}\\\\\\\implies\sf\:a\:-\:b\:\sqrt{7}\:=\:96\:\sqrt{7}\\\\\\\implies\sf\:a\:-\:b\:=\:\dfrac{96\:\cancel{\sqrt{7}}}{\cancel{\sqrt{7}}}\\\\\\\implies\sf\:a\:-\:b\:=\:96\\\\\\\implies\sf\:\sqrt{(\:a\:-\:b\:)}\:=\:\sqrt{96}\:\:\:-\:-\:[\:Taking\:square\:roots\:]\\\\\\\implies\sf\:\sqrt{(\:a\:-\:b\:)}\:=\:\sqrt{48\:\times\:2}\\\\\\\implies\sf\:\sqrt{(\:a\:-\:b\:)}\:=\:\sqrt{16\:\times\:3\:\times\:2}\\\\\\\implies\sf\:\sqrt{(\:a\:-\:b\:)}\:=\:4\:\sqrt{3\:\times\:2}\\\\\\\implies\boxed{\red{\sf\:\sqrt{(\:a\:-\:b\:)}\:=\:4\:\sqrt{6}}}$