To find any power of a given number.

nd the third power of 24.

-ION.-243 = 24 × 24 × 24 = 13824.

hat is the value of 632? Of 488? Of 324? Of 125? mon fraction is raised to any power by involving each

-Take the given number as many times as a factor as e units in the exponent of the power.

ution may sometimes be facilitated by means of the

CIPLE. The sum of the exponents of two powers of the ember is equal to the exponent of the product of those

22 x 23-25; for 22-2 × 2, and 23-2 × 2 × 2; hence 22 × 23=2 × 2 2=25.

power of a number is produced by multiplying its square by Its first power, square, and cube? Its cube by its cube?

nd the squares and cubes of the numbers from 4 to 16

e.

nd the squares and cubes of, 3, 7, 11, and 24. nd the squares of 1.05, .08, 4.8, 2.36, and .007.

the required power of the following:

390. To find the square of a number in terms of its tens and units.

1. Find the square of 27 in terms of its tens and units.

EXPLANATION. - The product of 20+7 by 7 is 20 × 7+72, and the product of 20+7 by 20 is 202+20 x7; hence, 202 + 2 × 20 × 7 + 72, which is the sum of these partial products, is the square of 20+ 7 or 27.

PRINCIPLE.

27 =

27=

189 =

540 =

20+7

20+7

20 × 7+72

202+20 × 7

729 = 202+2 × 20 × 7 + 72

The square of a number consisting of tens and

units is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units.

Using t and u respectively to denote the tens and units of a number, we have the

FORMULA

+ 2 x t x u + u2 = (t + u)2.

GEOMETRICAL ILLUSTRATION.

Let ABCD be a square, each side of which is D 27 ft., and draw the lines as represented in the figure. It is evident that in the square ABCD 272 is equal to the sum of two squares, one of which is the square of the tens 202, the other the square of the units 72, together with two rectangles, each of whose areas is 20 × 7.

2. Find the square of 37.

20

7

A

27

t2 + 2 × t × u+u2 = 302+2 × 30 × 7+72 = 1369 = 372, PRIN

391. 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.

EXPLANATION.-The square of 25 is 202+2 × 20 × 5 +5o (390, PRIN.). Multiply this square by 20+5; the product is 203+3 × 202 × 5 +3 × 20 × 5o +53, which is the cube of 25.

PRINCIPLE. The cube of a number composed of tens and units consists of four parts: 1st, the cube of the tens; 2d, three times the square of the tens multiplied by the units; 3d, three times the tens multiplied by the square of the units; 4th, the cube of the units. Hence,

FORMULA: (+ u}3 = 13 + 3 × t2xu + 3 xtx u2 + u3.

3. Find the cube of 34 in terms of its tens and units. 4. Find the cube of 45 in terms of its tens and units.

FIG. 3.

The volume of the cube marked A, Fig. 1, is 203; the volume of each of the rectangular solids marked B is 20 × 20 × 5, or 20 x 5; the volume of each of the rectangular solids marked C, in Fig. 2, is 20 × 5 × 5, or 20 x 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.

Find the value of each of the following expressions.

392. The root of a number is either the numbe or one of its equal factors.

393. Evolution is the process of finding a ro

and is the converse of involution.

1. The radical sign is √, or a fractional exponent. nall figure called the index, written above the radical sign, de he root. A fractional exponent is sometimes used. Thus, 16 is ent to 1/16; 64 to 1/64.

numerator of the exponent indicates a power, and the denominaot. Thus, 83 is equivalent to 8.

5. The square root of a number is one of its two equal Thus, the square root of 81 is 9, since 9 x9= 81.

S.

NCIPLES.-I. The square of a number contains twice as _figures as the number, or twice as many less one. Thus,

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

er.

he number of figures in the number is odd, the left-hand period ntain only one figure.

6. To find the square root of a number.

Find the square root of 729.

t

น

f2 + 2 × t× u + u2 = 7.29 (20 + 7 = 27