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EXERCISES.

1. What is the square of 27 ?

Answer, 729. 2. What is the cube of 139 ?

Answer, 2685619. 3. What is the 4th power of 78 ?

Answer, 37015056. 4. What is the 5th power of 35 ?

Answer, 52521875. When Vulgar Fractions are to be involved.

Example.
What is the 3d power of i?
Numerator. Denominator.
3 Root. 4 Root.

RULE. 3

Raise both the numerator and 9 Square. 16 Square. denominator separately to the

required power. 27 Cube. 64 Cube.

3

4

16

Answer, it

EXERCISES. 5. What is the square of į?

Answer, 6. What is the cube of į?

Answer, 7. What is the 4th power of ??

Answer, 725 8. Required the square of .

Answer, 961 When Decimal Fractions are to be involved.

Example.
What is the 4th power of .037 ?

.037 Root.
.037

Z209

[blocks in formation]

9. What is the square of 1.52 ? 10. What is the cube of .54 ? 11. What is the 4th power of .013 ? 12. What is the square of 82.17 ?

Answer, 2.3104.

Answer, .157464. Answer, .000000028561.

Answer, 6751.9089.

EVOLUTION.

Evolution, or the Extraction of Roots, is the method of finding the Square Root, Cube Root, &c. of a given number; being the reverse of Involution.

Roots are generally indicated by the radical sign » before the numbers : thus, N25, implies that the square root of 25 is to be extracted; 27, the cube root 27, &c. ; but sometimes they are expressed by a fraction placed over the number thus, 25|l; 27].

When no exact root of any number can be found, it is called a Surd Root.

To EXTRACT THE SQUARE ROOT OF A NUMBER.

RULE.

Place a point, or some other mark, over the right-hand or units figure, and every other figure to the left.

Find the nearest root in the first period, and place it on the right-hand, as a quotient, and the square under the said period.

Subtract the square, and bring down the next two figures to the remainder for a dividend.

Double the quotient for a divisor, and seek how often it is contained in the dividend, omitting the units place, and put the figure supposed, in the quotient, and also in the units place of the divisor.

Multiply the divisor by the figure last put in the quotient, and subtract the product from the dividend.

Bring down the next two figures, find divisor as before, repeating the foregoing directions to the end.

Point over the last figure, 6, and every

other to the left. Then the nearest root Example.

in the first period, 12, is 3. Put 3 in the

quotient; and 3 squared=9, which put What is the square root of 125316? under 12. Subtract, and 3 remain, to

which bring down the next two figures, 125316 (354

53: consequently 353 is a new dividend.

Double the quotient, 3, which is 6, and 9

place 6 for a divisor, leaving the units

place. 6 in 35, 5, (with a remainder.) 64) 353

Put 5 in the quotient, and also in the 325

units place of the divisor. Multiply the divisor, 65, by which is 325.

Subtract 325 from 353, and 28 remain. 704) 2816

Bring down the next two figures, 16; 2816

consequently 2816 is a new dividend. Double the quotient, 35, which is 70. Put 70 for a divisor, leaving the units place; 70 in 281 are 4. Put 4 in the quotient, and also in the units place of

the divisor. Multiply the divisor, 704, Answer, 354 square root.

by 4, which is 2816, the same as the

dividend. Example. What is the square root of Point over 6, and every other figure 5307705316 ?

to the left. The nearest root in the first

period, 53, is 7. Put 7 in the quotient: 5307705316 (72854

and 7 squared=49. 49 from 53, 4 re49

main, to which bring down the next two figures, 07; consequently, 407 is a new

dividend. Double the quotient, 7, 142) 407

which is 14. Put 14 for a divisor, leaving 284

the units place. 14 in 40, twice (with a remainder). Put 2 in the quotient, and

also in the units place of the divisor. 1448) 12370

Multiply the divisor, 142, by 2. Sub11584

tract 284 from 407, and 123 remain. Bring down the next two figures, 70.

Consequently, 12370 is a new dividend. 14565) 78653

Double the quotient 72=144. Put 144 72825

for a divisor; leaving the units place; 144 in 12370 are 8. Put 8 in the quotient

and also in the units place of the divi145704) 582816

sor. Multiply the divisor, 1448, by 8, 582816

which is 11584. Subtract 11584 from 12370, and 786 remain. And so by repeating the rule, we proceed to the end.

Proof of the last Example by Involution.

72854
72854 Root.

291416 364270 582832 145708 509978

5307705316 Square.

EXERCISES. 1. What is the square root of 625 ? 2. What is the square root of 59049 ? 3. What is the square root of 289444 ? 4. What is the square root of 2187441 ? 5. What is the square root of 6285049 ? 6. What is the square root of 5300568025 ?

Answer, 25. Answer, 243. Answer, 538. Answer, 1479. Answer, 2507. Answer, 72805.

When the given number has no exact square root, or is a Surd.

[blocks in formation]

7. What is the square root of 18?

Answer, 4.24264, &c. 8. What is the square root of 127 ?

Answer, 11.2694, &c. 9. What is the square root of 3857 ? Answer, 62.1047, &c. 10. What is the square root of 84276 ? Answer, 290.3032, &c.

When it is required to extract the square root of a number with decimals annexed.

[blocks in formation]

Answer, 19.4558, &c.
11. What is the square root of 19.0969 ?

Answer, 4.37. 12. What is the square root of 571.0463 ? Answer, 23.8965, &c. 13. What is the square root of 15.376 ? Answer, 3.92122, &c. 14. What is the square root of 28973.4? Answer, 170,215, &c. 15. Required the square root of 8 ? 16. Extract the square root of 53202436 ? 17. What is the square root of 703.28761 ? 18. Required the square root of 15241578750190521 ?

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