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305. TO FIND THE SQUARE Root of NUMBERS REPRE
SENTED BY LETTERS AND FIGURES.
Since the square root of a number is one of its two equal factors the square root of a*, (a x a x a x a), is a’, (a x a). The square root of a® is a. The square root of a is a'. Let a = 3, and verify each of the foregoing statements.
5. Let a = 2, b = 3, and c = 5, and find the numerical value of each of the following:
(a) Find the sum of the six results.
6. Let a =
2, 6 3, x = 5, and y = 7, and find the numerical value of each of the following:
(1) a v x*y? (2) Vx+y2 (3) a v x*yt
(5) 4 b Vyt (6) 5a V xijt
The factors of 4 a2b2 are 2, 2, a, a, b, b.
306. TO FIND THE Cube Root of NUMBERS REPRESENTED
BY LETTERS AND FIGURES.
EXPLANATION. Since the cube root of a number is one of its three equal factors, the cube root of a6, (a X a X a X a X a x a), is a?, (a X a). The cube root of a3 is a. The cube root of a' is a3. Let a= 2, and verify each of the foregoing statements.
1. y 16 = ?
3 53 = ?
37 = ?
2. ya 63 = ? VaRb3 = ? a 66 = ? Verify. 3. 3 8aR = ?
27 13 = ? 361a = ? Verify. 4. Let a = 2, b = 3, and c = 5, and find the numerical values of each of the following:
(1) y a 66
(2) 383 (3) y 27 63 (4) 2 3 a 63
(5) 3 3 633 (6) 43 68c3 (a) Find the sum of the six results.
307. MISCELLANEOUS PROBLEMS. Let a 2, b = 3, x = 5, and y = 7, and find the numerical value of each of the following:
(1) ab + Nvy* (2) abn xayt (3) 2a w x*y* (4) ab + V
(5) ab 3 x*y (6) 26 7 x®y (7) IV a'b2
(8) V ta'b2* (9) Nair? (b) Find the sum of the nine results.
* The factors of this number are }, }, a, a, a, a, b, 6.
308. THE SQUARE OF THE SUM OF TWO LINES.
1. Study the diagram and observe — D
E (1) That the line AC is the sum of
4 the lines AB and BC.
(2) That the square, 1, is the square of AB. (3) That the rectangle, 2, is as long
2 as AB and as wide as BC. (4) That the rectangle, 3, is as long
B as AB and as wide as BC.
(5) That the square, 4, is the square of BC.
(6) That the square, ACED, is the square of the sum of AB and BC.
2. Since a similar diagram may be drawn with any two lines as a base, the following general statement may be
The square of the sum of two lines is equivalent to the square of the first plus twice the rectangle of the two lines plus the square of the second.
3. If the line AB is 10 inches and the line BC, 5 inches, how
many square inches in each part of the diagram and how many in the sum of the parts ?
4. Consider the line AB 10 inches and the line BC 3 inches and find the area of each part of the diagram.
5. Suppose the line AB is equal to the line BC; what is the shape of 2 and 3?
6. In the light of the above diagram study the following:
14? = 196. (10 + 4)% = 10% + 2 (10 x 4) + 4* = 196.
309. MISCELLANEOUS REVIEW.
1. What is the square root of a' b??
What is the square root of 3 3 x 5 x 5 ?
2. What is the cube root of a 63 ?
What is the cube root of 2 x 2 x 2 x 7 x 7 x 7?
3. What is the square root of a' bh?
What is the square root of 52 x 347
4. What is the cube root of a® 68 ?
What is the cube root of 36 x 50 ?
5. The area of a certain square floor is 784 square
feet. How many feet in the perimeter of the floor?
6. The area of a certain square field is 40 acres. How many rods of fence will be required to enclose it ?
7. The solid content of a certain cube is 216 cubic inches. How many square inches in one of its faces ? ,
8. If there are 64 square inches in one face of a cube, how many cubic inches in its solid content ?
9. The square of (30 + 5) is how many more than the square of 30 plus the square of 5 ?
10. The square of (40 + 3) is how many more than the square of 40 plus the square of 3?
11. The square of a is aʼ; the square of 2 a is 4 a2. The square of two times a number is equal to how many times the square of the number itself?
12. The square of an 8-inch line equals how many times the square of a 4-inch line ?
13. The square of a 6-inch line equals how many times the square of a 2-inch line ?
SQUARE ROOT. 310. TO FIND THE Approximate Square Root of NUMBERS
THAT ARE NOT PERFECT SQUARES.
Find the square root of 1795. Regard the number as representing 1795 1-inch squares. These are to be arranged in the form of a square, and the length of its side noted.
100 l-inch squares = 1 10-inch square. 1700 1-inch squares
17 10-inch squares. But 16 of the 17 10-inch squares, can be arranged in a square that is 4 by 4; that is, 40 inches by 40 inches. See diagram.
After making this square (40 inches by 40 inches) there are (1700 — 1600 + 95) 195, 1-inch squares remaining. From these,
3 additions are to be made to two sides of the square already formed. Each side is 40
5 6 7 8 inches; hence the additions must be made upon a base line of 80 inches. These addi.
10 | 11 tions can be as many inches wide as 80 is contained times in 195. * 195 • 80 = 2 +.
13 14 | 15 16 The additions are 2 inches wide. These will require 2 times 80, + 2 times 2, = 164 square inches.
After making this square (42 in. by 42 in.) there are (195 – 164) 31 square inches remaining. If further additions are to be made to the square, the 31 square inches must be changed to tenth-inch squares. In each 1-inch square there are 100 tenth-inch squares; in 31 square inches there are 3100 tenth-inch squares. From these, additions are to be made upon two sides of the 42-inch square. 42 inches equal 420 tenth-inches. The additions must be made upon a base line (420 X 2) 840 tenth-inches long. These additions can be as many tenth-inches wide as 840 is contained times in 3100. 3100 = 840 = 3+. The additions are 3 tenth-inches wide. These will require 3 times 840, + 3 times 3, = 2329 tenth-inch squares.
After making this square (42.3 by 42.3) there are (3100 — 2529) 571 tenih-inch squares remaining. (If further additions are to be made to the square, the 571 tenth-inch squares must be changed to hundredthinch squares.)
The square root of 1795, true to tenths, is 42.3. * Allowance must be made for filling the little square shown at the upper right hand corner of the diagram.