Question

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

Answer 1

ANSWER:

Given:

• a = (√5-√3)/(√5+√3)
• b = (√5+√3)/(√5-√3)

To Find:

• a² + b²

Solution:

$\text{We are given that,}\\\\:\longrightarrow a=\dfrac{\sqrt5-\sqrt3}{\sqrt5+\sqrt3}\\\\\text{And,}\\\\:\longrightarrow b=\dfrac{\sqrt5+\sqrt3}{\sqrt5-\sqrt3} \\\\\text{Taking value of a and rationalising it,}\\\\:\implies a=\dfrac{\sqrt5-\sqrt3}{\sqrt5+\sqrt3}\\\\:\implies a=\dfrac{\sqrt5-\sqrt3}{\sqrt5+\sqrt3}\times\dfrac{\sqrt5-\sqrt3}{\sqrt5-\sqrt3}\\\\:\implies a=\dfrac{(\sqrt5-\sqrt3)(\sqrt5-\sqrt3)}{(\sqrt5+\sqrt3)(\sqrt5-\sqrt3)}\\\\\text{We know that,}\\\\:\hookrightarrow(x+y)(x-y)=x^2-y^2\\\\\text{So,}\\\\:\implies a=\dfrac{(\sqrt5-\sqrt3)^2}{(\sqrt5)^2-(\sqrt3)^2}\\\\\text{We know that,}\\\\:\hookrightarrow (x-y)^2=x^2-2xy+y^2$

$\text{So,}\\\\:\implies a=\dfrac{(\sqrt5)^2-2(\sqrt5)(\sqrt3)+(\sqrt3)^2}{5-3}\\\\:\implies a=\dfrac{5-2\sqrt{15}+3}{2}\\\\:\implies a=\dfrac{8-2\sqrt{15}}{2}\::\implies a=\dfrac{2\!\!\!/(4-\sqrt{15})}{2\!\!\!/}\\\\:\implies a=4-\sqrt{15}- - - -(1)\\\\\text{Now, rationalising b:}\\\\:\implies b=\dfrac{\sqrt5+\sqrt3}{\sqrt5-\sqrt3}\\\\:\implies b=\dfrac{\sqrt5+\sqrt3}{\sqrt5-\sqrt3}\times\dfrac{\sqrt5+\sqrt3}{\sqrt5+\sqrt3}\\\\:\implies b=\dfrac{(\sqrt5+\sqrt3)(\sqrt5+\sqrt3)}{(\sqrt5-\sqrt3)(\sqrt5+\sqrt3)}\\\\\text{We know that,}\\\\:\hookrightarrow(x+y)(x-y)=x^2-y^2\\\\\text{So,}$

$\implies b=\dfrac{(\sqrt5+\sqrt3)^2}{(\sqrt5)^2-(\sqrt3)^2}\\\\\text{We know that,}\\\\:\hookrightarrow (x+y)^2=x^2+2xy+y^2\\\\\text{So,}\\\\:\implies b=\dfrac{(\sqrt5)^2+2(\sqrt5)(\sqrt3)+(\sqrt3)^2}{5-3}\\\\:\implies b=\dfrac{5+2\sqrt{15}+3}{2}\\\\:\implies b=\dfrac{8+2\sqrt{15}}{2}\::\implies b=\dfrac{2\!\!\!/(4+\sqrt{15})}{2\!\!\!/}\\\\:\implies b=4+\sqrt{15}- - - -(2)$

$\text{We need to find,}\\\\:\longrightarrow a^2+b^2\\\\\text{We know that,}\\\\:\hookrightarrow x^2+y^2=(x+y)^2-2(x)(y)\\\\\text{So,}\\\\:\implies a^2+b^2\\\\:\implies(a+b)^2-2(a)(b)\\\\\text{From (1) and (2),}\\\\:\implies[(4-\sqrt{15})+(4+\sqrt{15})]^2-2(4-\sqrt{15})(4+\sqrt{15})\\\\:\implies(4-\sqrt{15}\!\!\!\!\!\!/\:\,+4+\sqrt{15}\!\!\!\!\!\!/\:\,)^2-2[(4)^2-(\sqrt{15})^2]\\\\:\implies(8)^2-2(16-15)\\\\:\implies64-2\\\\:\implies62\\\\\\bf{:\implies a^2+b^2=62}$

Formulae Used:

• (x + y)(x - y) = x² - y²
• (x + y)² = x² + 2xy + y²
• (x - y)² = x² - 2xy + y²
• x² + y² = (x + y)² - 2xy

Learn More:

$\boxed{\begin{minipage}{7 cm}\boxed{\bigstar\:\:\textbf{\textsf{Algebric\:Identities}}\:\bigstar}\\\\1)\bf\:(A+B)^{2} = A^{2} + 2AB + B^{2}\\\\2)\bf\: (A-B)^{2} = A^{2} - 2AB + B^{2}\\\\3)\bf\: A^{2} - B^{2} = (A+B)(A-B)\\\\4)\bf\: (A+B)^{2} = (A-B)^{2} + 4AB\\\\5)\bf\: (A-B)^{2} = (A+B)^{2} - 4AB\\\\6)\bf\: (A+B)^{3} = A^{3} + 3AB(A+B) + B^{3}\\\\7)\bf\:(A-B)^{3} = A^{3} - 3AB(A-B) - B^{3}\\\\8)\bf\: A^{3} + B^{3} = (A+B)(A^{2} - AB + B^{2})\\\\9)\bf\: A^{3} - B^{3} = (A-B)(A^{2} + AB + B^{2})\\\\ \end{minipage}}$