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1 June, 01:33

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

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  1. 1 June, 01:36
    0
    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)
  2. 1 June, 02:00
    0
    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.
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