3. Required the square root of 5678.243.

NOTE Here the decimal part contains an odd number of figures; therefore, annex a cipher, in order to complete the period in the decimal part, as follows:

5678,2430

4. Required the square root of 625.
5. Required the square root of 70225.
6. Required the square root of 194.
7. Required the square root of 2.
8. Required the square root of 998001.
9. Find the square root of 178857.
10. Find the square root of 75.347.

Ans. 75.354 +

Ans. 25. Áns. 265

Ans. 13.9283882 +
Ans. 1.4142135 +

Ans. 999.
Ans. 422.914.

Ans. 8.68026.

11. The square of a certain number is 944784; required the number.

Ans. 972. 12. What number is that whose square is 788544? Ans. 888. 13. A square field contains 106929 square yards. What the length of its side in feet? Ans. 981 feet.

14. If the area of a circle be 729 square yards; required the side of a square, of equal area in feet. Ans. 81

NOTE.-The following rule for extracting the square root in long calculations, will be found useful.

RULE. Find the square root by the above rule, to one figure more in the root than half the number of figures required. Take the next divisor for a common divisor to the remainder, annexing as many ciphers as may be required to complete the division, for the remaining part of the root see the following Example.

Required the square root of 5, to 10 places of decimals.

5(2,23606

4

42)100

84

443)1600
1329

4466)27100
26796

447206)3040000
2683236

447212)3567640(79775

3130484

4371560

4024908

3466520

3130484

3360360

3130484

2298760

2236060

62700

Therefore, the root is 2.2360679775.

CASE II-To extract the square root of a vulgar fraction.

But

RULE. The square root of a vulgar fraction may be found by taking that root of the numerator for a new numerator, and the root of the denominator for a new denominator. this rule cannot be strictly followed with advantage for when the root of a certain fraction is required, it frequently happens that it may be reduced either to lower or higher terms with advantage, or in many cases, it is advantageous to reduce the fraction to a decimal before we extract the root care being taken to have an even number of decimal places.

EXAMPLES.

1. Required the square root of 392.

Here; then / or (2) = 7.875 Ans.
Or thus:

= .765625; then /.765625 = .875. Ans.

2. Required the square root of

7

114

3. Required the square root of 25 4. Required the square root of 1. 5. Required the square root of 153. 6. Required the square root of 449. 7. Required the square root of 15.

8. Required the square root of

9. Required the square root of

10. Required the square root of

567

539°

Ans.

Ans. .94868

Ans. 3.9157 +
Ans. .8

Ans. 71428+

Ans. 531816 +

Ans. 12 or .583

Ans. .625

54

16

CUBE ROOT.

THE CUBE ROOT of a number is such a quantity, as when raised to its third power, is equal to the given number. Thus, the cube root of 8 is 2; for 2 X 2 X 2 = 8. The cube root of 1728 is 12; for 12 × 12 × 12 = 1728.

To find the cube root of any given number.

RULE Divide the given number into periods of three figures each, beginning at the units figure; as in the square root. If the number contain a decimal, the period must be made complete by annexing ciphers to the right hand of it.

The cube of your first period take,
And of its root a quotient make ;
Which root into a cube must grow,
And from your period taken fro'.
To the remainder, then you must
Bring down another period just,
Which being down, then you must see,
Your numbers straight divided be,
By just three hundred times the square
Of what your quotient figures are;
Which do so that you in may take,
The fact your quotient figures make.
Last squar'd and multipli'd by th' rest,
And product thirty times express'd.
The cube of your last found figure too,
You must put in, if right you do.
Repeat the work, and so descend
From point to point unto the end;
That done, if ought remain, you shall
Add triple ciphers for a decimal.

NOTE-In extracting the cube root of a decimal, care must be taken that the decimal places be three, or some multiple of three; because there are three times as many decimal places in the cube as in the root.

NOTE. Here the root is rational. When there is a remainder the root is irrational or a surd root.

The following rule may be used with advantage, particularly in long calculations.

1. Find the cube root by the common method, to one figure more in the root than half the figures required.

2. Divide the remainder with ciphers annexed, by three times the square of the figures in the root already found, for the remaining part of the root, see Example.