Question

5+2√3/5-2√3= a+b√3 find the value of and b

Answer 1

Answer:

a=5, b=2..............

Answer 2

$\huge\sf\pink{Answer}$

☞ A = 37/13

☞ B = 20/13

$\rule{110}1$

$\huge\sf\blue{Given}$

$\sf \dfrac{5 + 2 \sqrt{3} }{5 - 2 \sqrt{3} } = a + b \sqrt{3}$

$\rule{110}1$

$\huge\sf\gray{To \:Find}$

✭ Value of a and b?

$\rule{110}1$

$\huge\sf\purple{Steps}$

Consider LHS ;

$\leadsto \sf\dfrac{5 + 2 \sqrt{3} }{5 - 2 \sqrt{3} }$

To rationalise the denominator, we need to multiply numerator and denominator by the rationalising factor i.e., 5 + 2√3.

$\leadsto \sf\frac{5 + 2 \sqrt{3} }{5 - 2 \sqrt{3} } \\ \\ \leadsto \sf \frac{5 + 2 \sqrt{3} }{5 - 2 \sqrt{3} } \times \frac{5 + 2 \sqrt{3} }{5 + 2 \sqrt{3} } \\ \\ \leadsto \sf\frac{(5 + 2 \sqrt{3}) {}^{2} }{(5) {}^{2} - (2 \sqrt{3}) {}^{2} }$

[Since, (a + b)(a - b) = a² - b²]

$\dashrightarrow \sf \frac{(5) {}^{2} + (2 \sqrt{3}) {}^{2} + 2 \times 5 \times 2 \sqrt{3} }{25 - 12} \\ \\ \dashrightarrow \sf\frac{25 + 12 + 20 \sqrt{3} }{13} \\ \\ \dashrightarrow \sf \frac{37 + 20 \sqrt{3} }{13} \\ \\ \dashrightarrow \sf \frac{37}{13} + \frac{20}{13} \sqrt{3}$

On comparing LHS with RHS, we get -

$\large \color{aqua}{ \sf{\therefore a = \frac{37}{13} \: \: and\: \: b = \frac{20}{13}}}$

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