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
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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
Puzzle Answer
