3. Find the square of 42 in terms of its tens and units.

In like manner find the square

804. To find the cube of a number in terms of its tens and units.

1. Find the cube of 25 in terms of its tens and units.

253 = 203+ (3 × 202 × 5) + (3 × 20 × 52) +53

ANALYSIS.—The square of 25 is 202 + (2 × 20 +5)+5o. (803, PRIN.) Multiplying this by 20+5 gives the cube of 25.

2. Find the cube of 34 in terms of its tens and units.

PRINCIPLE.-The cube of a number consisting of tens and units is equal to the cube of the tens, plus three times the product of the square of the tens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units.

GEOMETRICAL ILLUSTRATION.

The volume of the cube marked A, Fig. 1, is 203; the volume of each of the rectangular solids marked B is 20 x 20 × 5, or 202 × 5; the volume of each of the rectangular solids marked C, in Fig. 2, is 20 × 5 × 5, or 20 × 52; and the volume of the small cube marked D is 53. It is evident, that if all these solids are put together as represented in Fig. 3, a cube will be formed, each edge of which is 25.

3. Find the cube of 46 ?

OPERATION.

408-64000

20

EVOLUTION

805. 1. What are the two equal factors of 25? 36? 2. What are the three equal factors of 27? 64? 125? 3. What are the four equal factors of 16? 81? 256? 4. Of what is 81 the 2d power? The 4th power?

DEFINITIONS.

806. The Square Root of a number is one of the two equal factors of that number; the Cube Root is one of the three equal factors of that number, etc.

Thus, 3 is the square root of 9, 2 is the cube root of 8, etc.

807. Evolution is the process of finding the root of any power of a number.

808. The Radical Sign is V. When prefixed to a number, it indicates that some root of it is to be found. 809. The Index of the root is a small figure placed above the radical sign to denote what root is to be found. When no index is written, the index 2 is understood.

3

Thus, 100 denotes the square root of 100; 125 denotes the cube root of 125; 256 denotes the fourth root of 256; and so on. Evolution, or both involution and evolution, may be indicated in the same expression by a fractional exponent, the numerator denoting the required power of the given number, and the denominator the root of that power of the number. Thus,

9 is equivalent to /9; 64, to /64; and 8, to the cube root of the second power of 8, equivalent to /83, etc.

EVOLUTION BY FACTORING.

WRITTEN EXERCISES.

810. To find any root of a number by factoring. 1. Find the cube root of 1728.

OPERATION.

3)1728

3)576

3)192

2)64

2)32

2)16

2)8

2)4

ANALYSIS.-A number that is a perfect cube, is composed of three equal factors, and one of them is the cube root of that number.

The prime factors of 1728 are 3, 3, 3, 2, 2, 2, 2, 2, 2; hence 1728 (3 × 2 × 2) × (3 × 2 × 2) × (3 × 2 × 2); therefore the cube root of 1728 is (3 × 2 × 2), or 12.

2

RULE.-Resolve the given number into its prime factors; then, to produce the square root, take one of every two equal factors; to produce the cube root take one of every three equal factors; and so on.

2. Find the square root of 64. Of 256. Of 576. Of 6561. 3. Find the cube root of 729. Of 2744. Of 9261. Of 3375.

GENERAL METHOD OF SQUARE ROOT.

811. A Perfect Square is a number which has an exact square root.

812. PRINCIPLES.-1. The square of a number expressed by a single figure contains no figure of a higher order than tens.

2. The square of tens contains no significant figure of a lower order than hundreds, nor of a higher order than thousands.

3. The square of a number contains twice as many figures as the number, or twice as many less one. Thus,

4. If any perfect square be separated into periods of two figures each, beginning with units' place, the number of periods will be equal to the number of figures in the square root of that number.

If the number of figures in the number is odd, the left-hand period will contain only one figure.

WRITTEN EXERCISES.

813. To find the square root of a number.

1. Find the square root of 4356.

OPERATION.

43,56 (60+6=66

3600

6C2= 120+6=126) 756

756

ANALYSIS.-Since 4356 consists of two periods, its square root will consist of two figures (812, PRIN. 4). Since 56 cannot be a part of the square of the tens (812, PRIN. 2), the tens of the root

square is contained in 4300 square of 6 tens, from the This remainder is composed

must be found from the first period 43. The greatest number of tens whose is 6. Subtracting 3600, which is the given number, the remainder is 756. of twice the product of the tens by the units, and the square of the units (803, PRIN.). But the product of tens by units cannot be of a lower order than tens; hence the last figure 6 cannot be a part of twice the product of the tens by the units; this double product must therefore be found in the part 750.

Now, if we double the tens of the root and divide 750 by the result, the quotient 6 will be the units' figure of the root, or a