9. 406 × 406 × 406 = 66923416; 406 is what root of 66923416?

10. What three equal factors will produce as their product?

11. What is the square root of ?

12. What is the cube root of 27?

13. If 4 is taken 3 times as a factor, what is the value of the power, and what root of the power is 4?

14. What power of 10 is 10?

15. What root of 5 is 5?

16. What is the first root of 261? 17. What is the second root of 1?

18. What is the cube root of 1?

19. 64 is the product of two equal factors; what are the factors?

20. 64 is the product of 6 equal factors; what is one of the factors?

SQUARE ROOT.

If we take the least and the greatest numbers that can be expressed by one, two, three, or any number of figures, we find that their squares will contain either twice as many, or one less than twice as many figures as the roots. Thus, 12 = 1, 92 = 81, 102 100, 992 = 9801, etc.

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From this we learn that there will be one figure in the root for every two figures in the second power, and an additional figure in the root for an odd figure in the power. Hence, in preparing a number for the extraction of the square root, we separate it into periods of two figures each, beginning at the right hand.

WRITTEN EXERCISES.

1. What is the square root of 225?

225(10+5

102=100

10 × 2=20)125 20+525 125

plus twice the product of the tens by the units, plus the square of the units in its root, we must find the square of the tens in the first left-hand period of the power. The greatest number of tens, whose square is contained in 2 hundreds, is 1. Subtracting the square of 1 ten, (102) = 100, from 225, there remains 125, which contains twice the product of the tens by the units of the root, plus the square of the units. Dividing 125 by 10 × 2 (twice the tens), we obtain 5 as the probable units' figure of the root. Adding the units' figure (5) to the trial divisor 20, we obtain the complete divisor 20+5. As this divisor consists of twice the product of the tens plus the units of the root, multiplying by the units' figure (5) and completing the division we obtain 125, which equals twice the product of the tens by the units (2 × 10 × 5) plus the square of the units (52); and the required root is 10+5=15.

RULE.

Separate the given number into periods of two figures each, beginning at the right hand. Find the greatest number whose square is contained in the first left-hand period, for the first figure of the root. Subtract the square of this figure from the left-hand period, and to the remainder annex the next period for a dividend.

Double the root already found, for a trial divisor, and find how often it is contained in the dividend, excluding the right-hand figure. Write the quotient as the next figure of the root, annexing it also to the trial divisor as the units' figure of the complete divisor. Multiply the complete divisor.

by the figure of the root just found, and subtract the product from the dividend.

Double the root already found for a new trial divisor, and continue the operation as before, till all the periods

have been used.

Note 1.-The left-hand period frequently contains but one figure.

Note 2.-If any trial divisor is not contained in its dividend, annex a cipher to the root, a cipher to the trial divisor, and another period to the dividend, and proceed as before.

Note 3.-If there is a remainder, after the root of a number has been extracted, annex periods of ciphers, and find the root to any required number of decimal places.

2. What is the square root of 2 to three decimal places?

PROOF.-Square the root 1.414 and add the remainder;

the result should be 2.

3. What is the square root of 961?

4. Extract the square root of each of the following numbers, 3025, 4356, and 6561.

5. What is the square root of 11881? 6. What is the value of 14016016? 7. Find the value of 1810162.0081.

Note.-The square root of a fraction can be found by extracting the square root of the numerator and of the denominator separately, or by changing it to a decimal and then extracting the root.

9. What is the value of each of the following: V 4

109

10. Extract the square root of .1 to three decimal places. 11. What is the value of 1.004 to three decimal places? 12. The area of a square lot is 1 acre; what is the length in yards of one of its equal sides?

13. What is one of the two equal factors of 15625? 14. I have a room in the form of a square, which requires 100 yards of carpet to cover it; what is the size of the room, if the carpet is 1 yard wide?

15. Find the side of a square whose area is 12 feet.

CUBE ROOT.

If we take the least and the greatest numbers that can be expressed by one, two, three, or any number of figures, we find that their cubes will contain either three times as many, or two or one less than three times as many figures as the roots. Thus, 13=1, 93=729, 103=10000, 993: 970299, 1003 1000000, 9993-997002999, etc.

=

=

From this we learn that there will always be one figure in the root for every three figures in the power, and an additional figure in the root if there is a partial period of either one or two figures after the full periods have been pointed off. Hence, in preparing a number for the extraction of the cube root, we separate it into periods of three figures each, beginning at the right hand.

times the square of the tens multiplied by the units, plus three times the square of the units multiplied by the tens, plus the cube of the units in its root, the cube of the tens must be found in the 3 thousands of the first period. The greatest number of tens whose cube is contained in 3 thousands is 1. Subtracting the cube of 1 ten, (103) = 1000, from 3375, there remains 2375, which contains three times the square of the tens multiplied by the units, three times the square of the units multiplied by the tens, and the cube of the units. Dividing 2375 by three times the square of the tens, or 102 × 3=300, as a trial divisor, the quotient is 7; but when we increase the trial divisor we find that 7 and 6 are both too great, and finally determine that 5 is the correct units' figure. We now add to the trial divisor three times the tens multiplied by the units (10 × 3 × 5) = 150, and the square of the units, (52)=25, making the complete divisor (102 × 3) + (10 × 3 × 5) + (52)=475. Multiplying by the units' figure we obtain 2375, and find that 10+5 or 15 the cube root of 3375.

If we multiply the complete divisor in parts, as in the example given in Involution, we shall have (102 × 3 × 5) + (10x3x5x5) +(52 × 5), an examination of which will show us that we have three times the square of the tens multiplied by the units, plus three times the tens multiplied by the square of the units, plus the cube of the units.

Note. As any number consisting of two or more figures may be divided into a number of tens and units, the analysis given above will apply, no matter how many figures there may be in the root.