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INVOLUTION.

Involution, or the raising of powers, is the method of finding the Square, Cube, &c. of any given number, and is performed by repeated multiplication.

Any number multiplied into itself produces the square, or second power; and that product, multiplied by the given number, produces the cube, or third power; and so on to any power whatever.

The number given to be involved, is called the Root, or first power.

The index, or exponent of a power, is the number which denotes the power: the square of 9 is often expressed 92; the cube of 8 as 83; the 4th power, or biquadrate of 23 as 234, &c.

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104

EXERCISES.

Answer, 729.

Answer, 2685619.

Answer, 37015056.

Answer, 52521875.

1. What is the square of 27? 2. What is the cube of 139?

3. What is the 4th power of 78?

4. What is the 5th power of 35?

When Vulgar Fractions are to be involved.

Example.

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Answer, 35 Answer,

7. What is the 4th power of ?

8. Required the square of 3.

When Decimal Fractions are to be involved.

Example.

What is the 4th power of . 037?

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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, 25, 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; 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 a divisor as before, repeating the foregoing directions to the end.

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Point over the last figure, 6, and every other to the left. Then the nearest root in the first period, 12, is 3. Put 3 in the quotient; and 3 squared=9, which put under 12. Subtract, and 3 remain, to which bring down the next two figures, 53: consequently 353 is a new dividend. Double the quotient, 3, which is 6, and place 6 for a divisor, leaving the units place. 6 in 35, 5, (with a remainder.) Put 5 in the quotient, and also in the units place of the divisor. Multiply the divisor, 65, by 5, which is 325. Subtract 325 from 353, and 28 remain. Bring down the next two figures, 16; 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, by 4, which is 2816, the same as the dividend.

Point over 6, and every other figure to the left. The nearest root in the first period, 53, is 7. Put 7 in the quotient: and 7 squared-49. 49 from 53, 4 remain, to which bring down the next two figures, 07; consequently, 407 is a new dividend. Double the quotient, 7, which is 14. Put 14 for a divisor, leaving the units place. 14 in 40, twice (with a remainder). Put 2 in the quotient, and also in the units place of the divisor. Multiply the divisor, 142, by 2. Subtract 284 from 407, and 123 remain. Bring down the next two figures, 70. Consequently, 12370 is a new dividend. Double the quotient 72-144. Put 144 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 divisor. Multiply the divisor, 1448, by 8, which is 11584. Subtract 11584 from 12370, and 786 remain. And so by repeating the rule, we proceed to the end.

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Answer, 538.

Answer, 1479.

Answer, 2507.

Answer, 72805.

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?

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

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