# Prove an Equation involving Square Roots

• Jun 10th 2011, 10:56 AM
BabyMilo
Prove an Equation involving Square Roots
Proof$\displaystyle \frac{a-1}{2 } \sqrt{\frac{a+1}{a-1 } } +\frac{a+1}{2 } \sqrt{\frac{a-1}{a+1 } } = \sqrt{a^2-1}$

fankyou.
• Jun 10th 2011, 11:08 AM
Quacky
Quote:

Originally Posted by BabyMilo
Proof$\displaystyle \frac{a-1}{2 } \sqrt{\frac{a+1}{a-1 } } +\frac{a+1}{2 } \sqrt{\frac{a-1}{a+1 } } = \sqrt{a^-1}$

fankyou.

I believe that there is a horrible typo in the question: it should be $\displaystyle \frac{a-1}{2 } \sqrt{\frac{a+1}{a-1 } } +\frac{a+1}{2 } \sqrt{\frac{a-1}{a+1 } } =\sqrt{a^2-1}$

I don't know if there is a shorthand, but I would consider writing the LHS as:
$\displaystyle \frac{1}{2}\sqrt{\frac{(a+1)(a-1)^2}{a-1}} +\frac{1}{2}\sqrt{\frac{(a-1)(a+1)^2}{a+1}}$

Take it from here.

Edit: I see you've corrected the typo in the OP
• Jun 10th 2011, 11:23 AM
BabyMilo
I was trying to correct it, but my internet is so slow.
Thank you for your solution :)
• Jun 10th 2011, 12:05 PM
Soroban
Hello, BabyMilo!

Another approach . . .

Quote:

$\displaystyle \text{Prove: }\;\frac{a-1}{2 } \sqrt{\frac{a+1}{a-1 } } +\frac{a+1}{2 } \sqrt{\frac{a-1}{a+1 } } \;=\; \sqrt{a^2-1}$

We have: .$\displaystyle \frac{a-1}{2}\cdot\frac{(a+1)^{\frac{1}{2}}}{(a-1)^{\frac{1}{2}}} + \frac{a+1}{2}\cdot\frac{(a-1)^{\frac{1}{2}}}{(a+1)^{\frac{1}{2}}}$

. . . . . . . $\displaystyle =\;\frac{(a-1)^{\frac{1}{2}}(a+1)^{\frac{1}{2}}}{2} + \frac{(a+1)^{\frac{1}{2}}(a-1)^{\frac{1}{2}}}{2}$

. . . . . . . $\displaystyle =\;\frac{[(a-1)(a+1)]^{\frac{1}{2}}}{2} + \frac{[(a+1)(a-1)]^{\frac{1}{2}}}{2}$

. . . . . . . $\displaystyle =\;\frac{(a^2-1)^{\frac{1}{2}}}{2} + \frac{(a^2-1)^{\frac{1}{2}}}{2}$

. . . . . . . $\displaystyle =\;(a^2-1)^{\frac{1}{2}}$

. . . . . . . $\displaystyle =\;\sqrt{a^2-1}$

• Jun 11th 2011, 12:04 PM
gsmani9
Equation involving square roots.
Quote:

Originally Posted by BabyMilo
Proof$\displaystyle \frac{a-1}{2 } \sqrt{\frac{a+1}{a-1 } } +\frac{a+1}{2 } \sqrt{\frac{a-1}{a+1 } } = \sqrt{a^2-1}$

fankyou.

L.H.S:-
$\displaystyle \frac{a-1}{2} \sqrt{\frac{a+1}{a-1}} + \frac{a+1}{2}\sqrt{\frac{a-1}{a+1}}$

$\displaystyle = \sqrt{\frac{(a+1)(a-1)(a-1)}{(a-1)4} } + \sqrt{\frac{(a-1)(a+1)(a+1)}{(a+1)4}}$

$\displaystyle = \sqrt{\frac{a^2-1}{4}} + \sqrt{\frac{a^2-1}{4}} = 2 \times \frac{1}{ 2}\sqrt{a^2-1}$

$\displaystyle = \sqrt{a^2 - 1}$

L.H.S. = R.H.S

I hope you have understood. All the best.