obtain-the-value-of-a-1-2-b-1-2-c-1-2-a-1-2-b-1-2-c-1-2-a-1-2-b-1-2-c-1-2-b-1-2-c-1-2-a-1-2-

Question Number 7610 by Tawakalitu. last updated on 06/Sep/16

o b t a i n t h e v a l u e o f

(a^{1 / 2} + b^{1 / 2} + c^{1 / 2}) (a^{1 / 2} - b^{1 / 2} + c^{1 / 2}) (a^{1 / 2} - b^{1 / 2} + c^{1 / 2}) (b^{1 / 2} + c^{1 / 2} - a^{1 / 2})

Commented by Rasheed Soomro last updated on 06/Sep/16

P e r h a p s y o u m e a n

(a^{1 / 2} + b^{1 / 2} + c^{1 / 2}) (a^{1 / 2} - b^{1 / 2} + c^{1 / 2}) (a^{1 / 2} + b^{1 / 2} - c^{1 / 2}) (b^{1 / 2} + c^{1 / 2} - a^{1 / 2})

Commented by Tawakalitu. last updated on 06/Sep/16

A l r i g h t s i r . . t h a n k y o u . i s l i k e t h a t .

Answered by Rasheed Soomro last updated on 06/Sep/16

(a^{1 / 2} + b^{1 / 2} + c^{1 / 2}) (a^{1 / 2} - b^{1 / 2} + c^{1 / 2}) (a^{1 / 2} + b^{1 / 2} - c^{1 / 2}) (b^{1 / 2} + c^{1 / 2} - a^{1 / 2})

L e t a^{1 / 2} = p, b^{1 / 2} = q, c^{1 / 2} = r

(p + q + r) (p - q + r) (p + q - r) (- p + q + r)

{(p + r) + q} {(p + r) - q} {q + (p - r)} {q - (p - r)}

{{(p + r)}^{2} - q^{2}} {q^{2} - {(p - r)}^{2}}

(p^{2} + 2 p r + r^{2} - q^{2}) (q^{2} - p^{2} + 2 p r - r^{2})

{2 p r + (p^{2} - q^{2} + r^{2})} {2 p r - (p^{2} - q^{2} + r^{2})}

{(2 p r)}^{2} - {(p^{2} - q^{2} + r^{2})}^{2}

4 p^{2} r^{2} - (p^{4} + q^{4} + r^{4} - 2 p^{2} q^{2} - 2 q^{2} r^{2} + 2 p^{2} r^{2})

2 p^{2} r^{2} - p^{4} - q^{4} - r^{4} + 2 p^{2} q^{2} + 2 q^{2} r^{2}

p = a^{1 / 2} \Rightarrow p^{2} = a \Rightarrow p^{4} = a^{2}

q = b^{1 / 2} \Rightarrow q^{2} = b \Rightarrow q^{4} = b^{2}

r = c^{1 / 2} \Rightarrow r^{2} = c \Rightarrow r^{4} = c^{2}

2 p^{2} r^{2} - p^{4} - q^{4} - r^{4} + 2 p^{2} q^{2} + 2 q^{2} r^{2}

= 2 a c - a^{2} - b^{2} - c^{2} + 2 a b + 2 bc

= - (a^{2} + b^{2} + c^{2} - 2 a b - 2 b c - 2 c a)

Commented by Tawakalitu. last updated on 06/Sep/16

w o w, t h a n k y o u s i r . G o d b l e s s y o u