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ber which being multiplied by itself once, will

produce the given number.

The cube root of a number is that number which being multiplied by itself twice, will produce the given number. For example, 6 is the square of root of 36: because 6x6=36; and 3 is the cube root of 27, because 3x3x3=27. The sign placed be fore a number denotes that its square root is to be extracted. Thus, V 36 =6. It is called the sign of the square root.

When we wish to express that the cube root is to be extracted, we place the figure 3 over the sign of the square root : thus, 3: V27=3.

EXTRACTION OF THE SQUARE ROOT.

$ 215. To extract the square root of a number, is to find a number, which being multiplied by itself

once will produce the given number. Thus vā= 2; for 2x2=4. And V9=3; for 3x3=9.

Roots 1. 2, 3, 4, 5, 6, 7, 8, 9.
Squares 1

4 9 16 25 36 49 64 81. From which we see that the square of either of the significant figures is less than 100, and hence the square root if any two figures will be less than 10. It is also evident that there are but nine perfect squares between 1 and 100.

CASE I. $ 216. To extract the square root of a whole number.

RULE. I. Point of the given number into periods of two figures each, counted from the right, by setting a dot over the place of units, another over the place of hun dreds and so on.

II. Find the greatest square in the first period on the left, and place its root on the right after the manner of a quotient in division. Subtract the square of the root from the first period, and to the remainder bring down the second period for a dividend.

III. Double the root already found and place it on the left for a divisor. Seek how many times the divisor is contained in the dividend, exclusive of the right hand figure, and place the figure in the root and also at the right of the divisor.

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

V. Double the whole root already found, for a new divisor, and continue the operation es before, until all the periods are brought down.

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EXAMPLES. 1. What is the square root of 263169 ? We first place a dot over

263169(513 the 9, making the right 25 hand period 69. We then

101)131 put a dot over the 1 and

101 also over the 6, making

1023)3069 three periods.

3069

The greatest perfect square in 26, is 25, the root of which is 5. Placing 5 in the root, subtracting its square from 26, and bringing down the next period 31, we have 131 for a dividend, and by doubling the ot we have 10 for a divisor. Now 10 is contain

13, 1 time. Place 1 both in the root and in

the divisor : then multiply 101 by 1; subtract the product and bring down the next period.

We must now double the whole root 51 for a new divisor, or we may take the first divisor after having doubled the last figure 1; then by dividing we obtain 3, the third figure of the root.

Note 1. There will be as many figures in the root as there are periods in the given number.

Note 2. If the given number has not an exact root, there will be a remainder after all the periods are brought down, in which case ciphers may be annexed, forming new periods, each of which will give one decimal place in the root. 2. What is the square root of 36729 ?

36729(191,64+. In this example

1 there are two pe

29,267
riods of decimals,

261
which
give two

381)629
places of decimals

381 in the root.

3826)24800

22956 38324)184400

153296

31104 Rem. 3. What is the square root of 106929 ?

Ans. 327. 4. What is the square root of 2268741 ?

Ans. 1506,23+. 5. What is the square root of 7596796 ?

Ans. 2756,22+. 6. What is tủe square root of 36372961 ?

Ans. 60

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7. What is the square root of 22071204 ?

Ans. 4698. CASE II. $ 217. To extract the square root of a decimal fraction.

RULE. I. Annex one cipher, if necessary, so that the number of decimal places shall be even.

II. Point of the decimal into periods of two fig. ures each, by putting a point over the place of tenths, a second over the place of thousandths, fc.: then extract the root as in whole numbers, recollecting that the number of decimal places in the root will be equal to the number of periods in the given decimal. Ex. 1. What is the square root of ,5?

,50(,707+

49 140) 100

000 1407)10000

9849

151 Rem. Note. When there is a decimal and a whole number joined together the same rule will apply. 2. What is the square root of 3271,4207 ?

Ans. 57,19+. 3. What is the square root of 4795,25731 ?

Ans. 69,247+. 4. What is the square root of 4,372594 ?

Ans. 2,091+. *Vhat is the square root of ,00032754?

Ans. ,01809+.

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CASE III. $ 218. To extract the square root of a vulgar i. tion.

RULE.

I. Reduce mixed numbers to improper fractions and compound fractions to simple ones, and then rea duce the fraction to its lowest terms.

II. Extract the square root of the numerator and denominator separately, if they have exact roots; but when they have not, reduce the fraction to a decimal and extract the root as in Case II. 1. What is the square root of 1.4 ?

Ans.. 2. What is the square root of 1781?

Ans. 3. What is the square root of mi?

Ans. 4.

4. What is the square root of 371 ?

Ans. ,89802+. 5. What is the square root of 197?

Ans. 86602+. 6. What is the square root of 171?

Ans. ,93309+

EXTRACTION OF THE CUBE ROOT.

$ 219. To extract the cube root of a number is to find a second number which being multiplied into itself twice, shall produce the given number.

Thus, 2 is the cube root of 8; for 2x2x2=8: and 3 is the cube root of 27 ; for 3x3x3=27. Roots 1 2, 3, 4, 5,

5, 6, 7, 8, 9. Cubes 1 8 27 64 125 216 343 512 729

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