Question

Rationlize the denomination 1 upon 3 √5 + 2 √2​

Answer 1

ANSWER :-

$\\ \bigstar \: \boxed{ \mathfrak{ \dfrac{1}{3 \sqrt{5} + 2 \sqrt{2} } = \dfrac{3 \sqrt{5} - 2 \sqrt{2}}{37} }}$

GIVEN :-

$\\ \implies \rm \: \frac{1}{3 \sqrt{5} + 2 \sqrt{2} }$

TO FIND :-

$\\ \implies \rm \: value \: of \: \: \frac{1}{3 \sqrt{5} + 2 \sqrt{2} }$

SOLUTION :-

$\\ \\ \implies \rm \: \dfrac{1}{3 \sqrt{5} + 2 \sqrt{2} } \\ \\ \\ \implies \rm \: \: \dfrac{1}{3 \sqrt{5} + 2 \sqrt{2} } \times \dfrac{3 \sqrt{5 } - 2 \sqrt{2} }{3 \sqrt{5} - 2 \sqrt{2} } \\ \\ \\ \implies \rm\: \dfrac{1 \big(3 \sqrt{5} - 2 \sqrt{2} \big)}{ \big(3 \sqrt{5 } + 2 \sqrt{2} \big) \big( 3 \sqrt{5} - 2 \sqrt{2} \big) } \\ \\ \\ \implies \rm\: \dfrac{3 \sqrt{5} - 2 \sqrt{2}}{(3 \sqrt{5} ) ^{2} - (2 \sqrt{2) ^{2} } } \\ \\ \\ \implies \rm \: \dfrac{3 \sqrt{5} - 2 \sqrt{2}}{9 \times 5 - 4 \times 2} \\ \\ \\ \implies \rm \: \dfrac{3 \sqrt{5} - 2 \sqrt{2}}{45 - 8} \\ \\ \\ \implies \boxed{\rm \: \dfrac{3 \sqrt{5} - 2 \sqrt{2}}{37} }$

→ Hence the answer is $\rm \dfrac{3 \sqrt{5} - 2 \sqrt{2}}{37}$

ADDITIONAL INFORMATION :-

→ ru/rs = rs. comman factor

→r/s + t/u = ru + st/su ......addition

→r/s - t/u = ru - st/su .......subtraction

→r/s x t/u = rt/su ........multiplication

→r/s ÷ t/u = ru/su. ........division

Answer 2

ANSWER✔

$\large\underline\bold{GIVEN,}$

$\sf\dashrightarrow \dfrac{1}{3\sqrt{5}+2 \sqrt{2}}$

$\large\underline\bold{TO,}$

$\sf\dashrightarrow\: RATIONALIZE\:THE\: DENOMINATOR.$

✯IDENTITY IN USE,

$\large{\boxed{\bf{ \star\:\: (a+b)(a-b)=a^2-b^2 \:\: \star}}}$

$\large\underline\bold{SOLUTION,}$

$\sf\star \dfrac{1}{3\sqrt{5}+2 \sqrt{2}}$

$\sf\red{MULTIPLYING \:BOTH \:NUMERATOR \:AND\: DENOMINATOR \:BY,\: 3\sqrt{5}-2 \sqrt{2} }$

WE GET,

$\sf\implies \dfrac{1}{3\sqrt{5}+2 \sqrt{2}}= \dfrac{3\sqrt{5}-2 \sqrt{2}}{3\sqrt{5}-2 \sqrt{2}}$

$\sf\implies \dfrac{3\sqrt{5}-2 \sqrt{2}}{ (3\sqrt{5}-2 \sqrt{2})(3\sqrt{5}+2 \sqrt{2})}$

$\sf\implies \dfrac{3\sqrt{5}-2 \sqrt{2}}{ (3\sqrt{5})^2-(2 \sqrt{2})^2} \:----$$\sf {\purple{\boxed{from\:given\:identity.}}}$

$\sf\implies \dfrac{3\sqrt{5}-2 \sqrt{2}}{ (9 \times 5)- (4 \times 2)}$

$\sf\implies \dfrac{ 3\sqrt{5}-2 \sqrt{2}}{ 45-8}$

$\sf\implies \dfrac{3\sqrt{5}-2 \sqrt{2}}{37}$

$\large{\boxed{\bf{\star\:\: \dfrac{3\sqrt{5}-2 \sqrt{2}}{37}\:\:\star }}}$

$\large\underline\bold\red{ THE\:RATIONALIZED\: DENOMINATOR\: OF\:\dfrac{1}{3\sqrt{5}+2 \sqrt{2}}\: IS\:\: \dfrac{3\sqrt{5}-2 \sqrt{2}}{37}}$

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