INVOLUTION

499. 1. Of what number are 3 and 3 the factors? 4 and 4?

2. Of what number are 3 and 3 and 3 the factors? 4 and 4 and 4?

3. What is the product when 5 is used twice as a factor? 4. What is the product or power, when 6 is used twice as a factor? When 8 is used twice as a factor?

6. What is the product when

When

is used twice as a factor?

is used three times as a factor?

7. What is the product of two 4's, or the second power of 4? What is the product of three 5's, or the third power of 5? What is the third power of 6?

8. What is the second power of? Of ? Of #?

DEFINITIONS.

500. A Power of a number is the product arising from using the number a certain number of times as a factor.

501. The powers of a number are named from the number of times the number is used as a factor.

Thus, when 2 is used as a factor twice, the product, 4, is called the second power of 2. 9 is the second power of 3. 27 is the third power of 3.

The number itself is called the first power.

502. The number of times a number is used as a factor is indicated by a small figure called an Exponent, written a little above and at the right of the number.

Thus, 32 means the second power of 3; 54, the fourth power of 5, etc. Inasmuch as the area of a square is the product of two equal factors, and the volume of a cube is the product of three equal factors, the second power of a number is also called the square, and the third power the cube of the number.

503. Involution is the process of finding the power of a number.

WRITTEN EXERCISES.

504. 1. Find the third power of 15.

PROCESS.

15X15X15=3375

ANALYSIS.-To find the third

power of a number is to find the product, when the number is used 3

times as a factor. Therefore, the third power of 15 will be 15×15 X15, which is equal to 3375.

21. Raise 10 to the fourth power; 8 to the third power; 3 to the 6th power.

505. To find the square of a number in terms of its

parts.

1. Find the square of 35 in terms of its tens and units.

ANALYSIS.-If we square 35 or multiply 35 by itself and write every step in the process, we shall have for the first product 25, or the square of the units, for the next two products 15 tens, or two times the product of the tens and units, and for the third product 9 hundreds or the square of the tens. Hence,

35

35

25=u2

}

15

PROCESS.

15

9

=2tX u

=t2

1225t2txu + u2

506. PRINCIPLE.-The square of any number consisting of tens and units, is equal to the tens2 + 2 times the tens the units +the units2.

Thus, 25=20+5, and 252 = 202 + 2 (20 × 5) + 5 2.

The above principle is true into whatever two parts the number may be separated, and the principle stated in general terms would be, the square of any number consisting of two parts is equal to the first part2 + 2 times the first párt X the second + second part2.

Thus, 148+6, and 142=82 + 2 (6 × 8) + 62.

Express in terms of their tens and units the square of the following numbers:

14. Square 16 by squaring its parts 9 and 7.
15. Square 20 by squaring its parts 12 and 8.
16. Square 32 by squaring its parts 30 and 2.
17. Square 13 by squaring its parts
18. Square 26 by squaring its parts
19. Square 17 by squaring its parts

11. 39.

12. 44

13. 67

7 and 6.

9 and 17.

8 and 9.

507. To find the cube of a number in terms of its

parts.

1. Find the cube of 35 in terms of its parts.

2. 26.

3. 31.

4. 28.

45

45

45

125

75

75

75

PROCESS.

27

5. 42.

6. 27.

7. 36.

=3t X u2

= t3

42875+3t2 × u + 3t × u2 + u3

ANALYSIS. By multiplying the second power expressed as in Art. 505, by 35, and writing every step, we shall have the cube of the tens, plus the product of three times the square of the tens multiplied by the units, plus the product of three times the tens multiplied by the square of the units, plus the cube of the units. Hence,

2

508. PRINCIPLE.-The cube of any number consisting of tens and units is equal to the tens 3+3 times the tens × the units +3 times the tens × the units' + the units3.

Thus, 25=20+5, and 253 = 203 + 3 (202 × 5) +3 (20 × 52) + 53.

The above principle may be stated in general terms thus: The cube of any number when separated into two parts is equal to the first part 3 +3 times the first part 2 second part + 3 times the first part multiplied by the second part 2 + the second part 3.

Express in terms of their tens and units the cube of the following numbers:

509. 1. What are the factors of 36? What are the two equal factors of 36? Of 49? Of 81?

2. What number used three times as a factor will produce 27? 64? 125? 216?

DEFINITIONS.

510. A Root of a number is one of the equal factors of the number.

Thus, 4 is a root of 16, because it is one of two equal factors.

Roots are named in a manner similar to powers. Thus, one of two equal factors of a number is the second, or square root; one of three equal factors, the third, or cube root; one of four equal factors, the fourth root, etc.

511. Evolution is the process of finding roots of numbers.

512. The Radical, or Root Sign, is √. When placed before a number it shows that its root is to be found.

When no figure or index is written in the opening of the radical sign, the square root is indicated; if the figure 3 is placed there, as, the cube root is indicated; if 4, as †, the fourth root; etc.

513. A Perfect Power is a number whose root can be found.