Anybody knows how to simplify ((10 + (108^.5))^(1 / 3)) - (((108^.5) - 10)^(1 / 3))?

((10 + (108^.5))^(1 / 3)) - (((108^.5) - 10)^(1 / 3))

= (10 + (108^(1/2))^(1/3) - ((108^(1/2)) - 10)^(1/3)

= (10 + 10.392)^(1/3) - (10.392 - 10)^(1/3)

= (20.392)^(1/3) - (.392)^(1/3)

= 2.732 - .732

= 2

You really want to do it by hand? Frankly, it will be quite clumsy but if it is an assignment, it will be desperately a piece of hard work. Yeh it is POSSIBLE to do it WITHOUT a calculator and... there you go.

* * *

To simplify the work, I'll set more variables to get rid of the clumsy numbers.

Let a = 108^.5 and b = 10;
so X will be (a+b)^(1/3)
& Y will be (a-b)^(1/3).

Let S be the expression,
i.e. S = X - Y
Taking cube on both sides, ...[trying to get rid of the nasty cubic roots]
　S^3
= (X-Y)^3 .............. [still remember binomial theorem?]
= X^3 - 3(X^2)Y + 3X(Y^2) - Y^3
= X^3 - Y^3 - 3XY(X-Y) ............[note that S = X - Y]
= (a+b) - (a-b) - 3[(a+b)(a-b)]^(1/3) x S
= 2b - 3[(a^2 - b^2)]^(1/3) x S
= 2 x 10 - 3(108-100)^(1/3) x S ......[8^(1/3) = 2]
= 20 - 6S

i.e. S^3 + 6S - 20 = 0

(S - 2)(S^2 + 2S + 10) = 0

For S^2+2S+10=0,
∆ = -36 < 0 i.e. no real solution

∴S = 2
The answer is 2.

108^.5 is 10.39230485

10.39230485 + 10 is 20.39230485

20.39230485^(1/3) is 2.732050808

Let A = 2.732050808

Now we do 10.39230485 - 10 is .3923048454

.3923048454^(1/3) is .7320508076

Let B = .7320508076

A - B =

2.732050808 - .732050808 = 2

simplified it's (20.39230485)^(1/3) - (.3923048454)^(1/3)

and the ones who got it right don't know what a vagina looks like.

pump it into your calculator.