Question

Find the value of a and b if
3 + √2 / 3-√ 2 = a+b √ 2


Please help

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:a\:=\:\dfrac{11}{7}}}}\sf\:\quad\:\&\:\quad\:\boxed{\red{\sf\:b\:=\:\dfrac{6}{7}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:\dfrac{3\:+\:\sqrt{2}}{3\:-\:\sqrt{2}}\:=\:a\:+\:b\:\sqrt{2}}$

We have to find the values of a and b.

Now,

$\displaystyle{\sf\:\dfrac{3\:+\:\sqrt{2}}{3\:-\:\sqrt{2}}\:=\:a\:+\:b\:\sqrt{2}}$

$\displaystyle{\implies\sf\:\dfrac{3\:+\:\sqrt{2}}{3\:-\:\sqrt{2}}\:\times\:\dfrac{3\:+\:\sqrt{2}}{3\:+\:\sqrt{2}}\:=\:a\:+\:b\:\sqrt{2}\:\quad\:\dots\:[\:Rationalising\:the\:denominator\:]}$

$\displaystyle{\implies\sf\:\dfrac{(\:3\:+\:\sqrt{2}\:)\:(\:3\:+\:\sqrt{2}\:)}{(\:3\:-\:\sqrt{2}\:)\:(\:3\:+\:\sqrt{2}\:)}\:=\:a\:+\:b\:\sqrt{2}}$

$\displaystyle{\implies\sf\:\dfrac{(\:3\:+\:\sqrt{2}\:)^2}{(\:3\:)^2\:-\:(\:\sqrt{2}\:)^2}\:=\:a\:+\:b\:\sqrt{2}\:\quad\:\dots\:[\:(\:a\:+\:b\:)\:(\:a\:-\:b\:)\:=\:a^2\:-\:b^2\:]}$

$\displaystyle{\implies\sf\:\dfrac{3^2\:+\:2\:\times\:3\:\times\:\sqrt{2}\:+\:(\:\sqrt{2}\:)^2}{9\:-\:2}\:=\:a\:+\:b\:\sqrt{2}\:\quad\:\dots\:[\:(\:a\:+\:b\:)^2\:=\:a^2\:+\:2ab\:+\:b^2\:]}$

$\displaystyle{\implies\sf\:\dfrac{9\:+\:2\:+\:6\:\sqrt{2}}{7}\:=\:a\:+\:b\:\sqrt{2}}$

$\displaystyle{\implies\sf\:\dfrac{11\:+\:6\:\sqrt{2}}{7}\:=\:a\:+\:b\:\sqrt{2}}$

$\displaystyle{\implies\sf\:\dfrac{11}{7}\:+\:\dfrac{6}{7}\:\sqrt{2}\:=\:a\:+\:b\:\sqrt{2}}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:a\:=\:\dfrac{11}{7}}}}\sf\:\quad\:\&\:\quad\:\underline{\boxed{\red{\sf\:b\:=\:\dfrac{6}{7}}}}\sf\:\quad\:\dots\:[\:By\:compairing\:]}$