As
logical continuation, the procedure to find a cube root without a
calculator or a personal computer is gointg to be analized and shown.
Talking
of operations, a cube root is a bit more difficult than a square
root. As a result,I would ask you for being patient.
So,
let’s go to use the number from the previous post, in other words
the number 18872162
in
order to find its cube root.
To
begin with, the next expression can be set out:
n3
=
(10a + b)3
+r
= 1000a3
+
300a2b
+ 30ab2
+ b3
+
r
Being
r the remainder of the cube root
Therefore,
the first term 1000a3
is
connecting with the first figure we obtain. So, the three central
terms from the previous expression are going to be used to find the
cube root:
300a2b
+ 300ab2
+ b3
which
will be added on the right of the cubic root:
The
procedure is quite similar to the one to solve a square root.
Next,
it is given details about:
1) The
figures are split up into groups of three
from right to left.
The
cube root is going to be solved with two decimal positions, so it
needs three ceros to obtain every decimal place.
2) The
first group on the left is taken which is 18. Then, a number that
cubed results 18 is looked for. This result must be as close as
possible to 18, by default, and it is substracted to 18.
The
number
is
2,
given
that
23=
8.
3)
The
following group of three figures is placed next
to 10, forming 10872.
4)
The
first term of the three listed on the definition at the beginning is
set.
The value that is placed between the parentheses is the one at this moment in the position of the solution, ie 2 ( 'a' in our definition ) :
300 x (2)2
x
[ ]
Now, 'b' that is the number
between square bracket is necessary to be known.
Removing two zeros the above
expression is:
3 x (2)2
x
[ ] = 12
x
[ ]
Then, 108 from 10872 is taken,
and is equalized to the previous expression
108 = 12
x
[ ]
The number inside the square
brackets is 9, which would be the next figure of the root.
However, here an obstacle can
come across as well as in a square root. This little problem is
related to the previous test so as to get an aproximate figure, due
to the fact that using this figure the result exceeds the number
which is going to be substracted. It can be seen in the next step.
4) After
the figure,which will be used to test, is obtained, the other two
central terms from the definition are proposed.
What
was pointed previously is possible to be seen here. That is why 16389
can not be substract to 10872, or colloquially, we have been through.
Now we test with 8.
Testing with 7 is required due
to the same problem.
And again the same. Thus the
figure has to be 6.
So, it
works now .
5) Afterwards,
the same procedure must be fulfilled again, placing the next group of
the decimal numbers next to 1296.
3 x 262 = 2028
¿How many 2028 are there in
12961? 6,391, so the test will be done using 6.
6) The process is fulfilled using
the other two groups of figures that remain in order to obtain the
cube root with two decimal figures.
All in all, the cube root of
18872162 with two decimal figures is 266.24 and the remainder would
be 75.701376. As it is possible to be seen, the more decimal figures
are added the more cumbersome the procedure becomes, but it think
that it can be understand properly.
I hope this has been
interesting and beneficial for you.
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