∛{7+5√2} + ∛{7-5√2} =?

1
by bhaiy
how did u type the cube root ? can uh elp me even i want to typw but i dont know
Comment has been deleted
1.99995

2015-10-01T16:24:32+05:30

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
Let  ∛(7 +5√2)    =  a  =  x ( 1 + y√2)
and  ∛(7 - 5√2)  = b

So  7 + 5√2  = x³ (1 + y √2)³
=  x³ (1 + 6 y²) + √2 * x³ (3 y+ 2 y³)
equate rational and irrational parts.
(1 + 6 y²) = 7 / x³    --- (1)
(3 y + 2 y³) = 5 / x³    ---- (2)

Solving these two equations we get:
14 y³ - 30 y² + 21 y - 5 = 0
looking at the coefficients :  14+21-30-5=0

so  y = 1 is a solution.  then  we get:  14 y² - 16 y + 5 as the other factor.    The roots of this polynomial are imaginary.

Thus we get y = 1.  hence by (1),    1 + 6 *1² =7/x³     =>  x = 1

so  ∛(7 + 5√2) =  1 + √2

similarly  ∛(7 - 5√2) = 1 - √2