Question

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

Answer 1

Answer:

         62

Step-by-step explanation:

One way to do this would be to work out and simplify each of a² and b², and then add them together and simplify again.

Another way, which avoids larger numbers, is to first work out and simplify a+b, and then use the fact that a²+b² = (a+b)² - 2ab.

First, notice that here b = 1/a, so ab = 1.

Next,

$\displaystyle a+b=\frac{\sqrt5-\sqrt3}{\sqrt5+\sqrt3}+\frac{\sqrt5+\sqrt3}{\sqrt5-\sqrt3}\\\\=\frac{(\sqrt5-\sqrt3)^2+(\sqrt5+\sqrt3)^2}{(\sqrt5+\sqrt3)(\sqrt5-\sqrt3)}\\\\=\frac{(5+3-2\sqrt{15})+(5+3+2\sqrt{15})}{5-3}\\\\=\frac{2\times(5+3)}{2}\\\\=8$

So...

a² + b²  =  (a+b)² - 2ab  =  8² - 2 = 64 - 2 = 62

Answer 2

Answer:

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