Place 1 in the root, also on the right of the 4 in the divisor. Multiply
41 by 1. Subtract and bring down the next period.
(4) 2 × 21 = 42. 42 is the _trial divisor_. 126 ÷ 42 = _about_ 3 times.
Place the 3 in the root also at the right of the 42 in the divisor.
Multiply out.
Square root = 213.
CUBE ROOT.--The _cube root_ of a number is one of the three equal
factors of that number.
Thus, 5 is the cube root of 125, because 5 × 5 × 5 = 125.
The _radical sign_ with a figure 3 over it (∛) means that the cube root
of the number following it is to be taken.
∛125 reads, “The cube root of 125.”
If we can find the prime factors of any perfect cube, we can write down
its cube root by inspection.
EXAMPLE: Find the cube root of 74088.
8|74088
+-----
9| 9261
+----- ∴ 74088 = 8 × 9 × 3 × 7 × 7 × 7
3| 1029 = 2³ × 3³ × 7³
+-----
7| 343 ∴ ∛74088 = 2 × 3 × 7
+----- = 42 _Ans._
7| 49
+-----
| 7
+-----
RULE FOR DIGITS.--Since 1³ = 1 and 10³ = 1000, therefore the cube of a
number which lies between 1 and 10 lies between 1 and 1000, _i. e._, the
cube of a number of one digit contains either one, two or three digits.
Again, since 10³ = 1000 and 100³ = 1000000, the cube of a number of two
digits contains either four, five, or six digits.
Proceeding in this way, we see that the cube of a number contains three
times, or one less or two less than three times, as many digits as the
number.
Hence, to find the number of digits in the cube root of a given number,
we mark off the digits in sets of three, beginning at the decimal point,
and marking both to the right and to the left.
[Illustration: =FIGURES REPRESENTING THE PROCESSES OF FINDING CUBE
ROOT=]
Thus, 289383 will be pointed off into two periods--289·383--and we
readily see there will be only 2 figures in the root.
The simplest method of finding the cube root of numbers whose prime
factors are not known is analogous to the method of finding square root,
being based upon the form of the cube of the sum of two numbers.
EXPLANATION: The cube root of 1728 is 12. Let us see how we found it.
12 = 1 ten + 2 units
12³ = (10 + 2)³
(10 + 2)³ means 10 + 2 × 10 + 2 × 10 + 2
10 + 2
10 + 2
--------------
10² + (10 × 2)
+ (10 × 2) +2²
--------------------
10² + 2(10 × 2) + 2²
10 + 2
-----------------------------
10³ + 2(10² × 2) + (10 × 2²)
+ (10² × 2) + 2(10 × 2²) + 2³
----------------------------------
10³ + 3(10² × 2) + 3(10 × 2²) + 2³