# Solving Cubic Roots By Hand

Using a method found here: https://www.wikihow.com/Calculate-Cube-Root-by-Hand, I was able to calculate cubic roots fairly quickly by hand. I did however run into something confusing. Most roots work fine but when I try to calculate the cubic root of 40, something goes wrong in the 3rd digit of the solution, I keep coming up with 9 instead of of the actual 1 I should be getting.

My math as follows:
$$3^3 = 27 <=40$$ Therefor first solution digit is 3.
$$40-27=13$$
Bringdown $$000 = 13,000$$
$$300*3^2=2700$$
$$2700*4=10,800<=13,000$$ Therefore next solution digit is "4".
$$3*10*3*4=360$$
$$4^2=16$$
$$Sum=3076$$
$$13,000-3076=9924$$ Bringdown $$000 = 9,924,000$$
This is where it goes wrong
$$300*34^2=346,800$$
$$346,800*9=3,121,200<=9,924,000$$ Therefore 3rd digit is 9 <-- WRONG (Should be 1)

Haven't gone further as that is wrong. The next two numbers are both 9, am I somehow skipping a line somewhere??

Any help would be appreciated.

ie. $$3076*4=12,304$$
$$13,000 - 12,304=696$$ bring down "000" etc.