Explain how you can approximate a cube root when the cube is not a perfect cube.

Step-by-step answer:

You can use Newton's method as follows:

It is an iterative method, meaning that from an approximation (i. e. an approximate solution), we can refine the answer to get a closer approximation. By repeating the process (iteration), we can get an answer as accurate as we wish.

The basis of the formula is based on

x1=x0-f (x0) / f' (x0)

where

x0 is a given approximation

x1 is a better approximation

f (x) is a given function (for which we need the roots)

f' (x) is the derivative of the function.

Skipping all details and applying directly to find the cube root, we have

N=the number for which the cube root is desired

f (x) = x^3-N

f' (x) = 3x^2

and x0 is an initial approximation that we need to provide (from the integer cubes, for example).

Say we need to find the cube-root of 124.

We know that 5^3 = 125, a rather close approximation, but the cube root is slightly less than 5.

To find a better approximation, we apply Newton's method, and calculate mentally:

x0=5, N=124

x1 = x0 - (x0^3-124) / (3 (x0^2)) [next substitute values]

= 5 - (125-124) / (3 (5^2)) [ next simplify ]

= 5-1/75 [next, rearrange to calculate mentally ]

=5 - (1/25) / 3 [ next, substitute 1/25 = 0.04, 1/3 division can be done mentally ]

= 5 - 0.04/3 [ divide 0.04/3 mentally ]

= 5 - 0.013333 [ subtract ]

= 4.98667

(exact value = 4.98663, first approximation is already quite close)

See below.

Step-by-step explanation:

It would take a long time to explain. There is a good method called Newton's method which involves graphs of the type

y = x^3 - n and applying calculus to produce cycles of approximation until you'll get close to the required cubic root.

You'll find it on online videos.