These expres

670. The second or square root of 4 is indicated thus, √4; the third or cube root of 8 is indicated thus, 8; the fourth root of 16, thus, 16; and so on. sions are read, "The square root of four"; "The cube root of eight"; "The fourth root of sixteen."

The symbol is called the radical sign. The small figure at the left of the radical sign is called the index of the root.

The root of a number is also indicated by a fractional exponent. Thus, 4 means the same as √√4; 8 means the same as $8.

SQUARE ROOT.

671. Oral Exercises.

a. What is one of the two equal factors of 25? of 49? b. What is the square root of 100? of 400? of 10000? of 1? c. What is the square root of ? of? of? of ?

d. What is the square root of 0.01? of 0.09? of 0.81?

To find the Number of Terms in the Square Root of a given

Number.

672. By squaring 1 and 9, 10 and 99, 100 and 999, etc., we obtain the following results:

These results show that when a number has one term its square is expressed by one or two figures; when a number has two terms the square is expressed by three or four figures; when a number has three terms the square is expressed by five or six figures, and so on. Hence,

673. If a numerical expression be separated into periods of two figures each, beginning with the units' figure, the number of periods will be the same as the number of terms in the square root.

To find the Parts of a Second Power.

674. To find what parts a second power is made up of, we may take a number consisting of tens and units, 36 for example, and raise it to the second power.

The written work above shows the partial products obtained by multiplying each term of the multiplicand by each term of the multiplier. From this we see that the second power of a number consisting of tens and units is made up of three parts, —

(1.) The square of the tens.

(2.) Twice the product of the tens by the units. (3.) The square of the units.

These parts may be expressed by the formula,

Tens2+ 2 (tens × units) + units2.

675. ILLUSTRATIVE EXAMPLE I. What is the square root of 1296 ? *

WRITTEN WORK.

Formula,

Tens2+2 (tens × units) + units2.

=

Explanation. We first find the number of terms in the root by separating the expression 1296 into periods 12/96 (36 of two figures each, beginning with the 9 units' figure. The square root of this number will consist of tens and units. We know that this number must have in it the square of the tens of the root, plus twice the product of the tens by the units, plus the square of the units.

(3 tens)2 = 3 tens x 2 = 6 tens) 39

36

36

36

* For another method of finding square roots, see Appendix, page 312.

As the first part of the power, the square of the tens, is hundreds, the 12 hundreds of the given number must have in it the square of the tens of the root.

The greatest square in 12 (hundreds) is 9 (hundreds), the square root of which is 3 (tens). This we write as the first term, or tens, of the root. Taking the square of 3 (tens), or 9 (hundreds), out of 12 (hundreds), there remain 3 (hundreds).

As the second part of the power, twice the product of the tens by the units, is tens, we unite the 9 tens of the given number with the 3 hundreds remaining, and have 39 tens.

This 39 tens has in it a product of which twice the tens of the root is one factor, and the units of the root the other factor. Dividing the 39 tens by twice 3 tens, 6 tens, we find 6 (units) to be the other factor, which we write as the second term or units of the root.

Multiplying the 6 (tens) by 6 (units), and taking the product 36 (tens) out of 39 (tens), we have 3 (tens) left.

As the third part of the power, the square of the units, is units, we unite the 6 units of the given number with the 3 tens remaining, and have 36 units. This 36 units has in it the square of the units of the root. Subtracting the square of 6 units, or 36, from 36, nothing remains.

Ans. 36.

676. ILLUSTRATIVE EXAMPLE II. What is the square. root of 1159.4025 ?

Explanation.-To extract

the square root of the integral part of the number 1159.4025, we proceed as in Illustrative Example I. Having found this part of the root, we consider it as tens in reference to the next term, double it for a new divisor, form a new dividend, and proceed as before.

So, whatever the number of terms in the root, having found a part of them, we consider the part found to

be the tens, double it for a new divisor, and proceed as before.

677. Rule.

To extract the square root of a number:

1. Beginning with the units' figure, point off the expression into periods of two figures each.

2 Find the greatest square in the number expressed by the left-hand period, and write its square root as the first term of the root.

3. Subtract this square from the part of the number used, and with the remainder unite the next term of the given number for a dividend.

4. Double the part of the root already found for a divisor, and by this divide the dividend, writing the quotient as the next term of the root.*

5. Multiply the divisor by this term, and subtract the product from the dividend. With the remainder unite the next term of the given number and subtract from the number thus formed the square of the term of the root last found.

6. If there are more terms of the root to be found, unite with the remainder the next term of the given number, take for a divisor double the part of the root now found, and proceed as before.

NOTE I. When, as in the work of Illustrative Example II., the divisor is not contained in the dividend, place a zero as the next figure in the root, place also a zero at the right of the divisor, and for a new dividend unite the next two terms of the given number with the previous dividend.

NOTE II. When there is a remainder after all the terms of the given number have been used, annex zeros to the remainder, and continue the work as far as desired.

NOTE III. The square root of a common fraction may be obtained by extracting the roots of both numerator and denominator, when they are perfect squares. When they are not perfect squares, first change the fraction to a decimal, and then extract its square root.

*If this quotient should prove to be too large, make it less and repeat the work.

678. Examples for the Slate.

Roots of numbers not perfect squares may be found to thousandths.

43. What is the length of one side of a square which con

tains 8836 square feet?

44. A body of troops consisting of 2401 men has an equal number in rank and file. How many are there in each ?

45. Find the side of a square that will contain as much surface as a rectangle 280 feet long and 70 feet wide.

46. Find the mean proportional between 42 and 168.

NOTE. The mean proportional between two numbers is the square root of their product. (See p. 256, Note I.)

47. Find the mean proportional between 56 and 224.

48. How many rods of fence will be required to enclose a square lot of 3 acres?

49. On one side of a roof there are laid 5000 slates, the number in the length being twice the number in the breadth. What is the number each way?

50. A and B each own a 10-acre lot. A's lot is square, and B's is twice as long as it is wide. How much greater length of fence will B require to enclose his lot than A to enclose his?

51. There is a rectangular court paved with 1728 pavingstones each 15 inches square; the length of the court is to the width as 4 to 3. What is the length and width of the court?

52. A rectangular block of granite is 8 ft. high, square at the base, and contains 162 cubic feet. What is the length of one side of the base?