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Square Root.
Note.--Pupils who have mastered the work on the preceding page will have no
difficulty in discovering that the same result may be obtained by the following
process:

Find the square root of 1795.
Operation.

Rule.
1795.'(42.36 1. Beginning with the
16

decimal point, group the fig-
40 x 2 = 80 * 195

ures as far as possible into 2 164

periods of two figures each.

2. Find the largest square 420 x 2 = 840 3100

3 2529 in the left-hand period and 4230 x 2 8460 57100

place its root at the right as 6 50796 the first figure of the com6304

plete root.

3. Subtract the square from the left-hand period and to the difference annex the next period. Regard this as a dividend.

4. Take 2 times 10 times the root already found as a trial divisor, and find how many times it is contained in the dividend. Write the quotient as the second figure of the root, and also as a part of the divisor. Multiply the entire divisor by the second figure of the root, subtract the product

from the dividend and proceed as before.

NOTE.-If, in applying the foregoing rule, a dividend is found that will pot contain the divisor, annex a zero to the root, a zero to the trial divisor, a new period to the dividend, and proceed as before.

PROBLEMS.
Find the approximate square root (true to tenths):

(1) 875. (2) 1526. (3) 2754. (4) 4150.
(5) 624. (6) 624.7

(7) 62.47. (8) 6.24.
(a) Find the sum of the eight results.

* The entire divisor is 80 and 2; that is, 82.
+ The left-hand period may consist of either one or two figures.

Square Root. 311. To FIND THE APPROXIMATE SQUARE ROOT OF DECI

MALS THAT ARE NOT PERFECT SQUARES.

Find the square root of .6 Regard the number as representing .6 of a 1-inch square. .6 of a 1-inch square= 60 tenth

inch squares.

But 49 of the 60 tenth-inch squares can be arranged in a square that is 7 by 7; that is, 7 tenths of an inch by 7 tenths of an inch.

After making this square there are (60 — 49) 11 tenth-inch squares remaining.

If additions are to be made to the square, the 11 tenth-inch squares must be changed to hundredth-inch squares. In each tenth-inch square there are 100 hundredth-inch squares ; in 11 tenth-inch squares there are 1100 hundredth-inch squares. From these, additions are to be made upon two sides of the .7-inch square.

.7 = 70 hundredths. The additions must be made upon a base line (70 X 2) 140 hundredth-inches long. These additions can be as many hundredth-inches wide as 140 is contained times in 1100. 1100 • 140 = 7+. The additions are 7 hundredthinches wide. These will require 7 times 140, +7 times 7, 1029 hundredth-inch squares.

After making this square (.77 by .77) there are (1100 — 1029) 71 hundredth-inch squares remaining. (If further additions are to be made to the square the 77 hundredth-inch squares must be changed to thousandth-inch squares.) The square root of .6, true to hundredths, is .77.

NOTE.—The work on this page should be first presented orally by the teacher. It must be given very slowly. Great care must be taken that pupils image each magnitude when its word-symbol is spoken by the teacher. Any attempt to move forward more rapidly than this can be done by the slowest pupil, will result in failure so far as that pupil is concerned. The great skill of many pupils in the Illinois School for the Blind in such work as this, is to be attributed mainly to their practice in creating imaginary magnitudes. When the teacher says tenth-inch square, they “think a tenth-inch square.” Its image comes immediately into consciousness. Teachers of pupils who have sight, may obtain invaluable suggestion from the mathematical ability of these pupils who have opportunity for comparatively little sense-perception of magnitudes, but who, from necessity, are constantly trained

see with the mind's eye." See preface of this book, page 3.

to"

Square Root. NOTE.—Pupils who have mastered the work on the preceding page will readily understand the following process. See rule on page 212.

1. Find the square root of .6.

.

Operation.

Observe460000.774

1. That in grouping deci49

mals for the purpose of extract

ing the square root it is 70 x 2 140 1100 7 | 1029

necessary to begin at the deci

mal point. 770 x 2 = 1540 7100 4 6176

2. That the square root of 924

any number of hundredths is a

number of tenths; the square root of any number of ten-thousandths is a number of hun. dredths, etc.

2. Find the square root of 54264.25.

54264. 25(232.946

4
20 x 2 40 142

3 129
230 x 2 =

460

1364

2 924 2320 x 2 4640 440.25

9 418.41 23290 x 2 46580 21.8400

4 18.6336 232940 x 2 = 465880 3.206400

6 2.795316

.411084 Observe that the trial divisor is always 2 times 10 times the part of the root already found.

Square Root.

312. The following numbers are perfect squares. their square roots by both the factor method and the method givou on the four preceding pages.

(2) 841

(1) 6889 (4) 1849

(3) 71824
(6) 60516

(5) 729

(a) Find the sum of the six results.

(7) 725

2 89

(10) 4489

(9) 82% (11) 2016

(12) 37 (b) Find the sum of the six results.

(13) .81
(14) .0625

(15) .04
(16) 1.21

(17) .7921 (18) .0004 (c) Find the sum of the six results.

or

or

or

313. MISCELLANEOUS. 1. The square of a number represented by one digit gives a number represented by

digits. 2. The square of a number represented by two digits gives a number represented by

digits. 3. The square of a number represented by three digits gives a number represented by

digits. 4. The square root of a perfect square represented by one or two digits is a number represented by

digit. 5. The square root of a perfect square represented by three or four digits is a number represented by digits.

6. The square foot of a perfect square represented by five or six digits is a number represented by

digits.

Square Root.

314. MISCELLANEOUS PROBLEMS.
1. What is one of the two equal factors of 9216 ?
2. What is one of the four equal factors of 20736 ?*

3. If a body of 7921 soldiers were arranged in a solid square, how many

soldiers would there be on each side? 4. How many rods of fence will enclose a square field whose area is 40 acres ?

5. How many rods long is one side of a square piece of land containing exactly one acre ? †

6. If the surface of a cubical block is 150 square inches, what is the length of one edge of the cube?

7. How many rods of fence will enclose a square piece of land containing 4 acres 144 square rods?

8. Find the side of a square equal in area to a rectangle that is 15 ft. by 60 ft.

9. Compare the amount of fence required to enclose two fields each containing 10 acres : one field is square, and the other is 50 rods long and rods wide.

10. Find the area of the largest possible rectangle having a perimeter of 40 feet.

11. If a square piece of land is of a square mile, how much fence will be required to enclose it?

12. Find the squares of numbers from 10 to 20, inclusive, and memorize them.

* To find one of the four equal factors of a number (the 4th root) extract the square root of the square root. Why? What is the fourth root of 81 ?

† Find the answer to problem 5, true to hundredths of a rod.

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