Approximate ∛-25 to the nearest tenth.

Even without a calculator, it is possible to estimate the cube-root of - 25 if you know a little about differential calculus and the Newton's rule, which says that given a close-enough approximation x0 to the cube-root of - 25, a better approximation x1 is given by

x1=x0 - (x0^3 - (-25)) / (3x0^2)

Say we have an initial approximation of x0=-3, since (-3) ^3=-27, not far from - 25.

A closer approximation is then

x1=-3 - ((-3) ^3 - (-25)) / (3 * (-3) ^2) = -3 - (-2/27) = -3+0.074=-2.926

which is accurate to the second digit without using a calculator, using all integer calculations and 2 one digit divisions.

So the final answer is - 2.9 (accurate to one decimal place)

(note 2/27 = (2/9) / 3 = 0.22222222 ... / 3=0.07407 ..., which can be handled without pencil and paper)

-2.92401773821 ≅ - 2.9