Powers and Roots.

304. MISCELLANEOUS PROBLEMS.

1. Square 42. Then resolve the square of 42 into its prime factors and compare them with the prime factors of 42. 2. Cube 42. Then resolve the cube of 42 into its prime factors and compare them with the prime factors of 42. 3. Square 45. Then resolve the square of 45 into its prime factors and compare them with the prime factors of 45.

4. Cube 45. Then resolve the cube of 45 into its prime factors and compare them with the prime factors of 45.

5. Divide the cube of 15 by the square of 15.

6. Divide the cube of by the square of 3.

7. Divide the cube of .7 by the square of .7.

8. Divide the cube of 2.5 by the square of 2.5.

9. Find the square root of 5 x 5 x 7 x 7.

10. Find the cube root of 3 × 3 × 3 × 5 × 5 × 5 × 7 x 7 x 7.

11. Find the square root of each of the following perfect

Algebra.

305. TO FIND THE SQUARE ROOT OF NUMBERS REPRESENTED BY LETTERS AND FIGURES.

. EXPLANATION.

Since the square root of a number is one of its two equal factors, the square root of a', (a × a × a × a), is a2, (a × a). The square root of a2 is a. The square root of a is a3. Let 3, and verify each of the foregoing statements.

a

3, x = 5, and y = 7, and find the numer

Algebra.

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 a, (a xaxaxaxaxa), is a2, (axa). The cube root of a3 is a. The cube root of a" is a3. Let a = 2, and verify each of the foregoing statements.

Geometry.

308. THE SQUARE OF THE SUM OF TWO LINES.

1. Study the diagram and observe—

(1) That the line AC is the sum of the lines AB and BC.

(2) That the square, 1, is the square

of AB.

(3) That the rectangle, 2, is as long as AB and as wide as BC.

D

(4) That the rectangle, 3, is as long A

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 made:

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. Suppose the line AB is equal to the line BC; what is the shape of 2 and 3?

5. In the light of the above diagram study the following:

142

= 196. (10 + 4)2 = 102+2 (10 × 4) + 42 = 196.

309. Miscellaneous Review.

1. What is the square root of a2b2?

What is the square root of 3 × 3 × 5 × 5 ?

2. What is the cube root of a3b3?

What is the cube root of 2 × 2 × 2 × 7 x 7 x 7 ?

3. What is the square root of a2b*?

What is the square root of 52 × 3a?

4. What is the cube root of ab?

What is the cube root of 36 × 5o?

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?

how

8. If there are 64 square inches in one face of a cube, 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 a2; the square of 2a is 4a2. 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?