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The Square Root of Numbers.

(a + h)2 = a2 + 2 a h + h2 = a2 + (2 a+ h) h it is unnecessary to find the square of the whole root at each successive approximation; for the square of a being already subtracted, it is sufficient to subtract (2a + h) h from the remainder, in order to obtain the next remainder. In this way, we obtain the following rule for the extraction of the square root.

To extract the square root of a number, divide it into periods of two figures each, beginning with the place of units.

Find the greatest square contained in the left hand period, and its root is the left hand figure of the required root.

Subtract the square of the root thus found from the left hand period, and to the remainder bring down the second period for a dividend.

Double the root for a divisor, and the quotient of the dividend exclusive of its right hand figure, divided by the divisor, is the next figure of the required root; which figure is also to be placed at the right of the divisor.

Multiply the divisor, thus augmented, by the last figure of the root, subtract the product from the dividend, and to the remainder bring down the next period for a new dividend.

Double the root now found for a new divisor and continue the operation as before, until all the periods are brought down.

The Square Root of Numbers.

190. EXAMPLES.

1. Find the square root of 28111204.

Solution. The operation is as follows:

281112045302 = Ans.

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4. Find the square root of 1607448649. Ans. 40093.

5. Find the square root of 48303584-206084.

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191. Corollary. The roots of vulgar fractions and mixed numbers may be computed in decimals by first reducing them to decimals.

The Square Root of Numbers.

192. EXAMPLES.

1. Find the square root of to 4 places of decimals. Ans. 0.2425+.

2. Find the square root of to 3 places of decimals. Ans. 0645+.

3. Find the square root of 1 to 2 places of decimals. Ans. 1.32+.

4. Find the square root of 111 to 3 places of decimals.

Ans. 3.418+

5. Find the 3d root of to 3 places of decimals.

Ans. 0.873+.

6. Find the 3d root of to 3 places of decimals.

Ans. 0.941.

7. Find the 3d root of 15 to 3 places of decimals.

Ans. 2.502+.

Power of a Monomial.

CHAPTER V.

POWERS AND ROOTS.

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SECTION I.

Powers and Roots of Monomials.

193. Problem. To find any power of a monomial.

Solution. The rule of art. 28, applied to this case, in which the factors are all equal, gives for the coefficient of the required power the same power of the given coefficient, and for the exponent of each letter the given exponent added to itself as many times as there are units in the exponent of the required power. Hence

Raise the coefficient of the given monomial to the required power; and multiply each exponent by the exponent of the required power.

194. Corollary. An even power of a negative quantity is, by art. 32, positive, and an odd power is negative.

195. EXAMPLES.

1. Find the third power of 2 a2 b5 c. Ans. 8a6 b15 c3.

2. Find the mth power of a".

3. Find the - mth power of a”.

4. Find the mth power of a-".

5. Find the -mth power of a ̃”.

Ans. am n

Ans. a-mn.

Ans. a-mn ̧

Ans. amn.

Root of a Monomial; imaginary quantity.

6. Find the 6th power of the 5th power of a3 b c2.

7. Find the qth power of the power of a-"

Ans. a90 630 60.

pth power of the mth Ans. amnpq.

8. Find the rth power of am b−n cr d.

Ans. amrb-nr cpr dr.

9. Find the 3d power of a-2b3 c−5 ƒ6 x−1.

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13. Find the 4th power of -36-3.

14. Find the 5th power of the 4th power of the 3d power

Ans 81b-12.

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196. To find any root of a monomial.

Solution. Reversing the rule of art. 193, we obtain immediately the following rule.

Extract the required root of the coefficient; and divide each exponent by the exponent of the required

root.

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