ANALYTICAL SOL.Squaring 25 by the method already given, we have 202+2× (5 × 20)+52. We then multiply this by 20+5. Five times 52 equals 53, 5 times 2 × 5 × 20

5 x 202 + 2 x 52 x 20+53

20+2 x 5 x 202 + 52 × 20 253203+3 x 5 x 202 + 3x 52 x 20 +53

equals 2 × 5 × 5 × 20, or 2 x 52 x 20, five times 202 equals 5 X 202. We next multiply by 20. Twenty times 52 equals 20 x 52, twenty times 2 x 5 x 20 equals 2 x 5 x 202, twenty times 202 equals 203. Taking the sum of these products and we have first 53; next, once 52 × 20 plus twice 52 X 20 equals three times 52 x 20; next twice 5 × 202 plus once 5 X 202 equals three times 5 X 202, and next we have 203; hence 203 + 3 x 5 x 202 + 3 x 52 x 20+53. Therefore the cube of 25 equals the cube of the tens, plus three times the square of the tens into the units, plus three times the tens into the square of the units, plus the cube of the units.

258

2. Find the cube of 45 by means of the cubical blocks.

GEOMETRICAL SOL.-Let A, Fig. 1, represent a cube whose sides are 40 units, its contents will be 403 = 64000. To increase its dimensions by 5 units we must add, 1st, the three rectangular slabs, B, C, D, Fig. 2; 2d, the three corner pieces, E, F, G, Fig. 3; The three 3d, the little cube H, Fig. 4.

slabs B, C, D, are 40 units long and wide and

OPERATION.

403 = 64000 402 × 5 × 3 = 24000 40 x 52 x 3 = 3000 53= 125

Hence 453

24000; the cou

5 units thick; hence their contents are 402 × 5 × 3 tents of the corner pieces, E, F, G, Fig. 3, whose length is 40 and breadth and thickness 5, equal 40 x 52 x 33000; and the contents of the little cube H, Fig. 4, equal 53= 125; hence the contents of the cube rep resented by Fig. 4 are 64000+24000+3000+125 91125

=

NOTE. When there are three figures in the number, complete the second cube as above, and then make additions and complete the third in the same manner; or let the first cube represent the cube already found, and then proceed as at first.

621. The following principles are important, and should be committed to memory:

PRINCIPLES..

1. The cube of a number consisting of two figures equals TENS3+3 times TENS2 X UNITS + 3 times TENS X UNITS 2+ UNITS3.

2. The cube of a number consisting of three figures equals HUNDREDS 3+3 times HUNDREDS2 × TENS + 3 times HUNDREDS TENS2 + TENS3 + 3 times (HUNDREDS + TENS) 2 × UNITS + 3 times (HUNDREDS + TENS) × UNITS2 + UNITS3. 622. These principles may also be expressed in symbols as follows:

3:

(t+u)3 = t3 +3t2.u+3t.u2+u3 (h+t+u)3=h3+3h2.t+3h.t2+ t3 + 3(h+t)2.u+ 3(h+t).u2 + u3.

EVOLUTION.

623. Evolution is the process of finding a root of a number.

624. A Root of a number is one of its equal factors. Roots are of different degrees; as, second, third, etc.

625. The Square Root, or second root, of a number is one of its two equal factors. Thus, 8 is the square root of 64, since 8×8 = 64.

==

626. The Cube Root, or third root, of a number is one of its three equal factors. Thus, 4 is the cube root of 64, since 4x4x4 = 64.

627. The Fourth Root, is one of the four equal factors; the fifth root is one of the five equal factors; etc.

628. The Symbol of Evolution is ✔; thus, ✅64 or ✓64, denotes the square root of 64; 3/64 denotes the cube root of 64.

629. The Index of the root is a small figure placed in the angle of the symbol. The index indicates the degree of the root.

Roots are also indicated by the denominator of a fractional exponent; thus 92 denotes 9; 27 denotes 27, etc.

630. The following principles of involution are given to enable us to determine the number of figures in the root:

PRINCIPLES.

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

12=1 9281 102 = 100

992= 9801

DEM.-The square of 1 is 1, and the square of 9 is 81, hence the square of a number consisting of one figure is a number consisting of one or two figures. The square of 10, the smallest number of two figures, is 100, the square of 99, the largest number of two figures, is 9801, hence the square of a number consisting of two figures is a number consisting of three or four figures, that is, twice two, or twice two less one, etc. The same may be shown for the square of a number consisting of any number of figures.

2. The cube of a number contains three times as many figures as the number itself, or three times as many, less one or two.

13 1 93729 103

993

1000

970299

DEM.-The cube of 1 is 1, and the cube of 9 is 729, hence the cube of any number consisting of one figure is a number consisting of one, two, or three figures. The cube of 10 is 1000, a number of four figures, the cube of 99 is 970299, a number of six figures, hence the cube of a number consisting of two figures contain four, five, or six figures, that is, three times two, or three times two less one The same may be shown for the cube of a number consisting of any number of figures.

or two.

EVOLUTION BY FACTORING.

631. When the number is a perfect power and the factors are easily found, the root of a number can be readily obtained by the following

Rule.-Resolve the number into its prime factors, and for the square root form a product by taking ONE of every Two equal factors; for the cube root ONE of every THREK equal factors; etc.

EXAMPLES FOR PRACTICE.

1. Find the square root of 144.

SOLUTION.-We first resolve the number into its prime factors. Since the square ot of a number is one of its two equal factors, we take one of every two equal factors and have 2×2× 3=12. Hence the square root of 144 is 12.

NOTE. We have marked the factors taken with a little star, and it will be well for the student to do the same in his solutions

Solve the following problems:

OPERATION.

2)144

*2)72

2)36

*2)18

3)9

*3

1. The square of a number is 64; what is the number?

2. of the square of a number equals 27; what is the number?

3. The square of twice a number equals 64;

4. The square of of a number equals 100;

what is the number? what is the number?

5. The square of twice a number is 18 more than twice the square of the number; what is the number?

6. Twice the square of a number is 8 more than 6 times the square of half the number; what is the number?

7. Fifteen is 3 more than of the cube of a number; what is that number?

8.cf the cube of a number is 10 more than the cube of of the number; what is the number?

9. The square of a number divided by the numbe equals 8; what is that number?

10. The square of a number divided by of the number equals 12 what is the number?

11. The cube of a number divided by the number equals 36; what is the number?

12. The 4th power of a number divided by the square of the numDer equals 49; what is the number?

13. The square of a number divided by of the number equals 27; what is the number?

14. A number divided by 6 gives double the square root of the number; what is the number?

15. The square of a number multiplied by one half of the number equals 32; what is the number?

16.of of the square of a number, multiplied by of of the number, equals 4; what is the number?

SQUARE ROOT.

632. There are Two Methods of explaining the general process of extracting the square root, called the Analytic or Algebraic Method, and the Synthetic or Geometrical Method.

633. The Analytic Method of square root is so called because it analyzes the number into its elements and derives the process of evolution from the law of involution.

634. The Geometrical Method is so called because it makes use of a geometrical figure to explain the process of extracting the root.

NOTE.-With young pupils who have a difficulty in understanding evolution it will be well to drill them upon the method of doing the work, not requiring them to give the explanation until they are better prepared to understand it.

1. Extract the square root of 1225.

ANALYTICAL SOLUTION.-Since the square of a number contains twice as many figures as the number itself, or twice as many less one, the square root of 1225 will consist of two places, and hence will consist of tens and units, and 1225 consists of tens2 + 2x tens × units+ units2.

OPERATION.

t2+2t. u+u2
t2=302

2t. u+u2=

2t

1225 (30 900 5

325 35

30 x 260 (2t+u). u⇒(60+5) × 5=325

The greatest number of tens whose square is contained in 1225 is 3 tens; squaring the tens and subtracting, we have 325, which equals 2 x tens X units+units. Now, since 2 X tens units must be greater than units2, 325 must consist principally of twice the tens into the units, hence if we divide by twice the tens we can ascertain the units. Twice the tens equals 30 × 2= 60; dividing, we find the units to be 5; now finding 2 × tens × units + units2, or, what is the same, 2 × tens units, both multiplied by units, which equals (60+5) × 5=325, and subtracting, nothing remains Fence the square root of 1225 is 3 tens and 5 units, or 35.