CUBES AND CUBE ROOTS
Cubes and cube roots are read on the K and D
scales of the slide rule. On the K scale are compressed
three complete logarithmic scales in
the same space as that of the D scale. Thus, any
logarithm on the K scale is three times the logarithm
opposite it on the D scale. To cube a
number by logarithms, we multiply its logarithm by three. Therefore, the
logarithms of cubed numbers will lie on
the K scale opposite the numbers on the
D scale.
As with the other slide rule scales mentioned,
the numbers the logarithms represent, rather
than the logarithmic notations, are printed
on the rule. In the lefthand third of the K
scale, the numbers range from 1 to 10; in the
middle third they range from 10 to 100; and in
the righthand third, they range from 100 to 1,000.
To cube a number, find the number on the D scale,
place the hairline over it, and read the digit
sequence of the cubed number on the K scale
under the hairline.
Placing the Decimal Point
The decimal point of a cubed whole or mixed number
may be easily placed by application of the
following rules:
1. If the cubed number is located in the left third
of the K scale, its number of digits to the left
of the decimal point is 3 times the number of
digits to the left of the decimal point in the original
number, less 2.
2. If the cubed number is located in the middle
third of the K scale, its number of digits is
3 times the number of digits of the original number,
less 1.
3. If the cubed number is located in the right
third of the K scale, its number of digits is
3 times the number of digits of the original number.
EXAMPLE: (1.6)^{3}
SOLUTION : Place the hairline over 16 on D scale.
Read the digit sequence, 409, on the K scale
under the hairline.
Number of digits to left of decimal point in the number
1.6 is 1 and the cubed number is in the lefthand
third of the K scale.
3 x (No. of digits)2 = (3 x 1)2
= 1
Therefore,
(1.6)^{3} = 4.09
EXAMPLE: (4.1)^{3}
Digit sequence = 689.
SOLUTION: Number of digits to left of decimal
point in the number 4.1 is 1, and the cubed number
is in the middle third of the K scale.
3 x (No. of digits)1 = (3 x 1)1
= 2
Therefore,
(4.1)^{3}
= 68.9
EXAMPLE: (52)^{3}
SOLUTION: Digit sequence = 141.
Number of digits to left of decimal point in the number
52 is 2, and the cubed number is in the righthand
third of the K scale.
3 x No. of digits = 3 x 2
= 6
Therefore,
(52)^{3} = 141,000
Positive Numbers Less Than One
If positive numbers less than one are to be cubed,
count the zeros between the decimal point and
the first nonzero digit. Consider the count
negative. Then the number of zeros between the decimal point and the first
significant digit of the cubed number may be
found as follows:
1. Left third of K scale: Multiply the zeros counted
by 3 and subtract 2.
2. Middle third of K scale: Multiply the zeros
counted by 3 and subtract 1.
3. Right third of K scale: Multiply the zeros counted
by 3.
EXAMPLE: Cube 0.034
SOLUTION: Digit sequence = 393
Zero count of 0.034 = 1, and 393 is in the middle third of the K scale.
3 x (No. of zeros) 1 = (3 x 1)1 = 4
Therefore,
(0.034)^{3}
= 0.0000393
Practice problems. Cube the following numbers using the slide rule.
1. 21
2. 0.7
3. 0.0128
4. 404
Answers:
1. 9260
2. 0.342
3. 0.0000021
4. 66,000,000
Cube Roots
Taking the cube root of a number on the slide
rule is the inverse process of cubing a number.
To take the cube root of a number, find the
number on the K scale, set the hairline over
it, and read the cube root on the D scale under
the hairline.
