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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 left-hand 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 right-hand 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 left-hand 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 right-hand 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.




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