The task: ** evaluate
{(√50-√49)}^(1/3) **

may be accomplished by several methods, inter alia by:

Pure algebra
Calculus
Calculator
Trial and error
Computer programs
Other
Please solve by at least 2 distinct ways and compare results.

(V50-V49)^(1/3) = (5V2 -7)^(1/3) =Vx - y(say)

Then, we must have:

xVx - 3*x*y + 3*Vx*y^2- y^3 = 5^2 -7

=> Vx(x+3*y^2) - (3*x*y +y^3) = 5V2 -7

Then, we have:

3xy + y^3 =7 ...........(i)

and, Vx(x+ 3*y^2) = 5*V2.......(ii)

Now, (x, y) = (2,1) satisfy eqn (i)

Substituting: (x,y) = (2,1) in (ii), it also satisfies the equation.

Therefore, (V50-V49)^(1/3) = V2 - 1

This is corroborated by determining the calculator value of (V50-V49)^(1/3), which is approximately equal to: 0.4142135623731, which is also equal to (V2-1).

Trial and error

---------------------

By way of simple calculator, we observe that:

5*V2 -7 =~ 0.07106781186

Now, 0.41^3=0.068921

and, 0.42^3 = 0.074088

Similarly, in this way, (5V2-7)^(1/3) =~ 0.4142135 = V2-1

*Edited on ***January 18, 2024, 10:16 am**