In raising a number to the third power, a combination of the preceding
method and ordinary multiplication is employed.
TO FIND THE CUBE OF A NUMBER.—_Set the_ L.H. _or_ R.H. _index of C to
the number on D, and opposite the number_ ON THE LEFT-HAND _scale of B
read the cube on the_ L.H. _or_ R.H. _scale of A_.
By this rule four scales are brought into requisition. Of these, the D
scale and the L.H. B scale are _always_ employed, and are to be read as
of equal denomination. The values assigned to the L.H. and R.H. scales
of A will be apparent from the following considerations.
Commencing with the indices of C and D coinciding, and moving the slide
to the right, it will be seen that, working in accordance with the above
rule, the cubes of numbers from 1 to 2·154 (= ∛(10)) will be found on
the first or L.H. scale of A. Moving the slide still farther to the
right, we obtain _on the_ R.H. _A scale_ cubes of numbers from 2·154 to
4·641 (or ∛(10) to ∛(100)). Had we a _third_ repetition of the L.H. A
scale, the L.H. index of C could be still further traversed to the
right, and the cubes of numbers from 4·641 to 10 read off on this
prolongation of A. But the same end can be attained by making use of the
R.H. index of C, when, traversing the slide to the right as before, the
cubes of numbers from 4·641 to 10 on D can be read off _on the_ L.H. _A
scale_ over the corresponding numbers on the L.H. B scale. Hence, using
the L.H. index of C, the readings on the L.H. A scale may be regarded
comparatively as units, those on the R.H. A scale as tens; while for the
hundreds we again make use of the L.H. A scale in conjunction with the
_right-hand_ index of C.
By keeping these points in view, the number of digits in the cube (N) of
a given number (_n_) are readily deduced. Thus, if the units scale is
used, N = 3_n_ − 2; if the tens scale, N = 3_n_ − 1; while if the
hundreds scale be used, N = 3_n_. Placed in the form of rules:—
N = 3_n_ − 2 when the product is read on the L.H. scale of A with the
slide to the _right_ (units scale).
N = 3_n_ − 1 when the product is read on the R.H. scale of A; slide to
the _right_ (tens scale).
N = 3_n_ when the product is read on the L.H. scale of A with the slide
to the _left_ (hundreds scale).
With decimals the same rule applies, but, as before, the number of
digits must be read as −1, −2, etc., when one, two, etc., cyphers follow
immediately after the decimal point.
EX.—Find the value of 1·4^3.
Placing the L.H. index of C to 1·4 on D, the reading on A opposite 1·4
on the L.H. scale of B is found to be about 2·745 [2·744].
EX.—Find the value of 26·4^3.
Placing the L.H. index of C to 26·4 on D, the reading on A opposite 26·4
on the L.H. scale of B is found to be about 18,400 [18,399·744].
EX.—Find the value of 7·3^3.
In this case it becomes necessary to use the R.H. index of C, which is
set to 7·3 on D, when opposite 7·3 on the L.H. scale of B is read 389
[389·017] on A.
EX.—Find the value of 0·073^3.
From the setting as before it is seen that the number of digits in the
number must be multiplied by 3. Hence, as there is −1 digit in 0·073,
there will be −3 in the cube, which is therefore read 0·000389.
The last two examples serve to illustrate the principle of factorising
with powers of 10. Thus
0·073 = 7·3 × 10^{−2}; 0·073^3 = 7·3^3 × (10^{−2})^3 = 389 × 10^{−6} =
0·000389.
_Cube Root_ (_Direct Method_).—One method of extracting the cube root of
a number is by an inversion of the foregoing operation. Using the same
scales, _the slide is moved either to the right or left until under the
given number on A is found a number on the_ L.H. _B scale, identical
with the number simultaneously found on D under the right or left index
of C_. This number is the required cube root.
From what has already been said regarding the combined use of these
scales in cubing, it will be evident that in extracting the cube root of
a number, it is necessary, in order to decide which scales are to be
used, to know the number of figures to be dealt with. We therefore (as
in the arithmetical method of extraction) point off the given number
into sections of three figures each, commencing at the decimal point,
and proceeding to the left for numbers greater than unity, and to the
right for numbers less than unity. Then if the first section of figures
on the left consists of—
1 figure, the number will evidently require to be taken on what we have
called the “units” scale—_i.e._, on the L.H. scale of A, using the L.H.
index of C.
If of 2 figures, the number will be taken on the “tens” scale—_i.e._, on
the R.H. scale of A, using the L.H. index of C.
If of 3 figures, the number will be taken on the “hundreds”
scale—_i.e._, on the L.H. scale of A, using the R.H. index of C.
To determine the number of digits in cube roots it is only necessary to
note that when the number is pointed off into sections as directed,
there will be one figure in the root for every section into which the
number is so divided, whether the _first_ section consists of 1, 2, or 3
digits.
Of numbers wholly decimal, the cube roots will be decimal, and for every
group of _three_ 0s immediately following the decimal point, _one_ 0
will follow the decimal point in the root. If necessary, 0s must be
added so as to make up complete multiples of 3 figures before proceeding
to extract the root. Thus 0·8 is to be regarded as 0·800, and 0·00008 as
0·000080 in extracting cube roots.