Question

ques - If a and B are rational numbers and 3√5+√3/√5-√3 = a+b√15 find the values of a and b​

Answer 1

$\sf\red{\underline{\underline{Answer:}}}$

$\sf{The \ value \ of \ a \ and \ b \ are \ 10 \ and \ 2}$

$\sf{respectively.}$

$\sf\orange{Given:}$

$\sf{\implies{a+b\sqrt15=\frac{3\sqrt5+\sqrt3}{\sqrt5-\sqrt3}}}$

$\sf\pink{To \ find:}$

$\sf{The \ value \ of \ a \ and \ b.}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{\implies{a+b\sqrt15=\frac{3\sqrt5+\sqrt3}{\sqrt5-\sqrt3}}}$

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$\sf{\implies{\frac{3\sqrt5+\sqrt3}{\sqrt5-\sqrt3}}}$

$\sf{On \ rationalising \ the \ denominator}$

$\sf{\implies{\frac{3\sqrt5+\sqrt5(\sqrt5+\sqrt3)}{\sqrt5-\sqrt3(\sqrt5+\sqrt3)}}}$

$\sf{\implies{\frac{15+3\sqrt15+5+\sqrt15}{\sqrt5^{2}-\sqrt3^{2}}}}$

$\sf{\implies{\frac{20+4\sqrt15}{5-3}}}$

$\sf{\implies{\frac{2(10+2\sqrt15)}{2}}}$

$\sf{\implies{10+2\sqrt15}}$

__________________________________

$\sf{\therefore{a+b\sqrt15=10+2\sqrt15}}$

$\sf{On \ comparing}$

$\boxed{\sf{a=10}}$

$\sf{b\sqrt15=2\sqrt15}$

$\sf{\therefore{b=\frac{2\sqrt15}{\sqrt15}}}$

$\boxed{\sf{\therefore{b=2}}}$

$\sf\purple{\tt{\therefore{The \ value \ of \ a \ and \ b \ are \ 10 \ and \ 2}}}$

$\sf\purple{\tt{respectively.}}$