how to find the cube root of a number Aug 22 2014 The Newton Raphson method approximates the roots of a function So we need a function whose root is the cube root we re trying to calculate Let s say we re trying to find the cube root of 3 And let s say that x is the cube root of 3 Therefore x3 3 For the Newton Raphson method to be able to work its magic we need to set
Cube root of 15 correct to three decimal places is x 2 466 Let the cube root of 15 be x Then cubing both sides and interchanging x 3 15 x 3 15 0 Let f x x 3 15 We need to investigate for the roots of the polynomial x 3 15 Consider any perfect cube close to the number 15 1 3 1 2 3 8 3 3 27 15 lies between 8 and 27 Hence cube root of 15 lies You divide the exponent by 3 For any real number a it is true that root 3 a a 1 3 So if a is written as power a b c then to calculate the cube root you have to divide the power s exponent by 3 root 3 b c b c 1 3 b 1 3c b c 3
how to find the cube root of a number
how to find the cube root of a number
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The cube roots of 8 are 2 2omega and 2omega 2 where omega 1 2 sqrt 3 2 i is the primitive Complex cube root of 1 Here are the cube roots of 8 plotted in the Complex plane on the circle of radius 2 graph x 2 y 2 4 x 2 2 y 2 0 01 x 1 2 y sqrt 3 2 0 01 x 1 2 y sqrt 3 2 0 01 0 5 5 2 5 2 5 They can be written as 2 cos 0 i It is any complex number z which satisfies the following equation z n 1 where n in NN which is to say that n is a natural number A natural number is any positive integer n 1 2 3 This is sometimes referred to as a counting number and the notation for it is NN
Yes you can Example Evaluate root 3 125 We can write 125 as the product of 3 negative 5 s Hence root 3 125 5 The reason that we can t have the square or quartic root of a negative number is that two negatives always cancel to give a plus so the square root of a negative number is undefined For example sqrt 64 sqrt 8 xx 8 so sqrt 64 is Group the decimals in three s from the decimal Each group must have one number 3 0 000 343 3 0 000 0 3 343 7 3 0 000 343 0 07 Check 2 decimal places cubed will give 6 decimal places Answer link root 3 0 000343 0 07 Note that 0 000343 343 10 6 and 343 7 3 So root 3 0 000343 root 3 343 10 6 root
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Now cubic roots of 125 2 1 3i are given as 125e 2 3 i 1 3 5 ei 2k 2 3 1 3 5ei 3 2k 2 3 5ei 2k 3 2 9 Where k 0 1 2 Thus setting k 0 1 2 in above general cubic root we get three cubic roots of given complex number 3 125 2 1 3i 5e 2 i 9 5e 4 i 9 5e 10 i 9 This is an example of Casus Irreducibilis a Complex cube root that is not expressible in the form a bi where a and b are expressed in terms of Real n th roots You can express it in terms of trigonometric functions z itself lies on the unit circle z cos pi 6 i sin pi 6 Use De Moivre s formula
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