SOLUTION: How to determine the cubic root step by step without calculator?

Question 579307: How to determine the cubic root step by step without calculator?

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you can get close by first thinking of an integer that would get above it when cubed and another integer that would get below it when cubed.

example:
suppose you wanted to find the cube root of 100.

4 cubed = 64
5 cubed = 125

your number will be between 4 and 5.

split the difference and try 4.5
4.5 cubed equals 91.125 which is too low
try 4.6
4.6 cubed equals 97.336 which is still too low
try 4.7
4.7 cubed equals 103.823 which is too high.

your number will be between 4.6 and 4.7

split the difference and try 4.65
4.65 cubed equals 100.544625 which is too high.
try 4.64
4.64 cubed equals 99.897344 which is too low.

your number will be between 4.65 and 4.64

split the difference and try 4.645
4.645 cubed equals 100.2206361 which is too high.
try 4.644
4.644 cubed equals 100.155922 which is still too high.
try 4.643
4.643 cubed equals 100.0912357 which is still too high.
try 4.642
4.642 cubed equals 100.0265773 which is still too high.
try 4.641
4.641 cubed equals 99.96194672 which is too low.

your number will be between 4.641 and 4.642.

split the difference again and keep doing this until you get as close to 100 as you think is necessary.

obviously, the closer you get, the more complex the multiplication gets, but that's the nature of the beast. the calculator does it automatically by either having a table that it can look up or by using some algorithm that does the same thing very quickly. since the table would have to be extremely large to account for all the possibilities, the calculator is more then likely using some algorithm (formula) to arrive at the answer. the reason it can do it so quickly is a combination of a very efficient algorithm and the fact that the calculator takes millions of steps per second which is a lot faster than you can do it by hand.

the actual cube root of 100 by using the calculator comes out to be 4.641588834.

we didn't do too bad but we had a long way to go depending on the accuracy we were looking for.

4.641 cubed is equal to 99.96194672 which is 99.96% of 100 which is only .04% off the mark.
4.6415 cubed is equal to 99.99425852 which is 99.99% of 100 which is only .01% off the mark.
5 decimal point accuracy is not bad, but obviously we can do a lot better and a lot quicker through the use of a calculator.

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