Question

please solve it with steps I want I will mark you brainlist

if you span then you don't love your paranets it means ​

Answer 1

Solution:

Given,

$\tt\implies\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}=a+b\sqrt{15}$

We have to find out the values of a and b.

So,

$\tt\implies\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}=a+b\sqrt{15}$

Multiplying both numerator and denominator by (√5 + √3), we get,

$\tt\implies\dfrac{(\sqrt{5}+\sqrt{3})(\sqrt{5}+\sqrt{3})}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})}=a+b\sqrt{15}$

Using identity (a + b)(a - b) = a² - b², we get,

$\tt\implies\dfrac{(\sqrt{5}+\sqrt{3})^{2}}{(\sqrt{5})^{2}-(\sqrt{3})^{2}}=a+b\sqrt{15}$

$\tt\implies\dfrac{(\sqrt{5}+\sqrt{3})^{2}}{5-3}=a+b\sqrt{15}$

Using identity (a + b)² = a² + 2ab + b², we get,

$\tt\implies\dfrac{5+3+2\times\sqrt{5}\times\sqrt{3}}{2}=a+b\sqrt{15}$

$\tt\implies\dfrac{8+2\sqrt{15}}{2}=a+b\sqrt{15}$

$\tt\implies4+1\sqrt{15}=a+b\sqrt{15}$

Comparing both sides, we get,

$\begin{cases}\tt a=4\\ \tt b=1\end{cases}$

Answer:

• a = 4 and b = 1.

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