EXTRACTION OF THE SQUARE ROOT.

§ 180. 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 √93; for, 3×3=9.

Roots
Squares.

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

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 of any number expressed by two figures will be less than 10. It is also evident that there are but nine perfect squares between 1 and 100 among the whole numbers.

Q. What is required when we wish to extract the square root of a number? What is the greatest square of a single figure? Is the square of a single figure always less than 100? Will the square root of two figures be less than 10?

CASE I.

§ 181. 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. But if the product should exceed the dividend, diminish the last figure of the root.

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

EXAMPLES.

1. What is the square root of 263169? We first place a dot over the 9, making the right hand period 69. We then put a dot over the 1 and also over the 6, making three periods.

OPERATION.

26 31 69(513

25

101)131

101

1023)3069

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 root we have 10 for a divisor. Now 10 is contained in 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 ?

3. What is the square root of 106929? 4. What is the square root of 2268741?

Ans. 327.

Ans. 1506,23+.

5. What is the square root of 7596796 ?

Ans. 2756,22+.

6. What is the square root of 36372961? 7. What is the square root of 22071204?

Ans. 6031.

Ans. 4698.

Q. How do you extract the square root of a whole number? How many figures will there be in the root? If the given number has not an exact root, what may be done?

CASE II.

§ 182. 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 decimals into periods of two figures each, by putting a point over the place of hundredths, a second over the place of ten 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.

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+.

5. What is the square root of ,00032754?

Ans. ,01809+.

6. What is the square root of,00103041? Ans. ‚0321. 7. What is the square root of 4,426816? Ans. 2,104. 8. What is the square root of 47,692836? Ans. 6,906.

Q. How do you extract the square root of a decimal fraction? When there is a decimal and a whole number joined together, will the same rule apply?

CASE III.

§ 183. To extract the square root of a vulgar fraction.

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.

Q. How do you extract the square root of a vulgar fraction?

EXTRACTION OF THE CUBE ROOT.

184. 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.

1, 2, 3, 4, 5, 6, 7, 8, 9. 64 125 216 343 512 729.

Roots

Cubes

1 8

27

CASE 1.

§ 185. To extract the cube root of a whole number.

RULE.

I. Point off the given number into periods of three figures each, by placing a dot over the place of units, a second over the place of thousands, and so on to the left: the left hand period will often contain less than three places of figures.

II. Seek the greatest cube in the first period, and set its root on the right after the manner of a quotient in division. Subtract the cube of this figure from the first period, and to the remainder bring down the first figure of the next period, and call the number the dividend.

III. Take three times the square of the root just found for a divisor and see how often it is contained in the dividend" and place the quotient for a second figure of the root. Then cube the figures of the root thus found, and if their cube be greater than the first two periods of the given number, diminish the last figure, but if it be less, subtract it from the first two periods, and to the remainder bring down the first figure of the next period, for a new dividend.

IV. Take three times the square of the whole root for a new divisor, and seek how often it is contained in the new dividend: the quotient will be the third figure of the root. Cube the whole root and subtract the result from the first three periods of the given number, and proceed in a similar way for all the periods.