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 6×6=36; and 3 is the cube root of 27, because 3x3x3=27. The sign placed before a number denotes that its square root is to be extracted. Thus, √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, 3/27=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 √4= 2; for 2×2=4. And √9=3; for 3×3=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 off 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 hundreds 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 dividend, 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 as before, until all the periods are brought down.

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 have 131 for a dividend, and by doubling the ve 10 for a divisor. Now 10 is containtime. 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 ?

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 the square root of 36372961 ?
Ans. 60

8

7. What is the square root of 22071204 ?

CASE II.

Ans. 4698.

§ 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 off the decimal into periods of two figures each, by putting a point over the place of tenths, a second over the place of thousandths, &c.: 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+.

What is the square root of ,00032754?

Ans. ,01809+.

CASE III.

§ 218. To extract the square root of a vulgar à tion.

RULE.

I. Reduce mixed numbers to improper fractions and compound fractions to simple ones, and then reduce 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 294?

2. What is the square root of 2794 ?

3. What is the square root of lo?

12544

4. What is the square root of 1?

Ans. &.

Ans. 1.

Ans. .

Ans. ,89802+.

5. What is the square root of 247 ?

Ans. 86602+.

6. What is the square root of $78?

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 2×2×2=8: and 3 is the cube root of 27; for 3×3×3=27. Roots 1 2, 3, 4, 5, 6, 7, 8, 9. Cubes 1 8 27 64 125 216 343 512 729.