Question

Approximate ∛-25 to the nearest tenth.

Answer 1

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)

Answer 2

-2.92401773821 ≅ -2.9