Question

5+√3/5-√3=a+b√15 find the value of a and b

Answer 1

Answer:

a = 14, b = 5

Step-by-step explanation:

Given : $\dfrac{5 + \sqrt{3} }{5 - \sqrt{3} }$

Rationalising the denominator, we get

→ $\dfrac{5 + \sqrt{3} }{5 - \sqrt{3} } \times \dfrac{5 + \sqrt{3} }{5 + \sqrt{3} }$

→ $\dfrac{(5 + \sqrt{3} )(5 + \sqrt{3} )}{(5 - \sqrt{3})(5 + \sqrt{3} ) }$

{Identity : (a - b)(a + b) = a² - b²

Here, a = 5, b = √3}

→ $\dfrac{ {(5 + \sqrt{3} )}^{2} }{ {(5)}^{2} - {( \sqrt{3} )}^{2} }$

{Identity : (a + b)² = a² + b² + 2ab

Here, a = 5, b = √3}

→ $\dfrac{ {(5)}^{2} + {( \sqrt{3} )}^{2} + 2(5)( \sqrt{3} )}{5 - 3}$

→ $\dfrac{25 + 3 + 10 \sqrt{3} }{5 - 3}$

→ $\dfrac{28 + 10 \sqrt{3} }{2}$

→ $\dfrac{2(14 + 5 \sqrt{3}) }{2}$

→ 14 + 5√3

On comparing with a + b√3, we get

a = 14, b = 5