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dently be repeated; thus, we shall have ob and ba, ac and ca, &c. Therefore, the number of combinations will be.

10×9
2

If now we add an eleventh article, each of the eleven may be joined to each of the combinations of the remaining ten, and we 11 X 10×9 shall have permutations. But each combination will 1X2

be three times repeated; thus we shall have abc, bac, and cab; abd, bad, and dab, &c. The number of combinations of 3 out 11 X 10X9 of 11 will therefore be

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1x2x3

RULE.

Hence we obtain the fol

Write for a numerator the descending series, commencing with the number from which the combinations are to be made, and for a denominator the ascending series, commencing with 1, giving to each series as many terms as are equivalent to the number in one combination.

Cancel the like factors in the numerator and denominator, and divide.

10. How many combinations of 4 letters, can be made from the alphabet?

11. How many combinations of 7 can be made from 18 apples?

12. How many ranks of 10 men, may be made in a company of 80 ?

13. How many locks of different wards, may be unlocked with a key of 6 wards? [Find the number of combinations of 1, 2, 3, 4, 5, and 6 in 6, and the sum of all the combinations will be the number required.]

CHAPTER XV.

INVOLUTION.

INVOLUTION is the repeated multiplication of a number by itself.

The product obtained by Involution is called a power. The root is the number involved, or the first power. If the root be multiplied by itself, or employed twice as a factor, the product is the second power. If the root is

employed three times as a factor, it is raised to the 3d power; if 5 times, to the 5th power, &c. Thus, 2 is the 1st power of 2, or 21. 2×2 or 4, is the 2d power of 2, or 22. 2×2× 2 or 8, is the 3d power of 2, or 23. 2× 2 ×2×2×2 or 32, is the 5th power of 2, or 25. The power is usually denoted by a small figure over the right of the root, called the exponent, or index. When there is no exponent, the number is regarded as the 1st power.

The second power is often called the square, because the number of square feet in any square surface, is obtained by multiplying the number of feet in one side by itself.

The third power is often called the cube, because the number of cubic feet in any cubical block, may be ob tained by raising the number of feet in one side to the 3d power.

The 4th power is sometimes called the bi-quadrate, or the square squared; the 5th power, the first sursolid; the 6th power, the square cubed, or the cube squared; the 7th power, the second sursolid; the 8th power, the bi-quadrate squared; the 9th power, the cube cubed; the 10th power, the 1st sursolid squared, &c.

If the exponents of any two powers of the same number be added, we shall obtain the exponent of their product. Thus 63 × 65=6×6×6×6 × 6 × 6 × 6 × 6 = 68 ; 42 × 43-4X 4x4x4x4=45.

In any two powers of the same number, if we subtract the smaller exponent from the larger, we shall obtain their 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 quotient. Thus 6865.

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6 × 6 × 6 × 6 × 6

We may represent any power of a number, by multiplying its exponent. Thus, the 7th power of 5 is 57; the 3d power of 22 is 26, because 22 × 22 × 22-26. These properties form the basis of the system of Logarithms.

1. What is the 2d power of 6? the 3d power?
2. Find the value of .94; 123; (4)5; 29.

3. Find the value of 164; 1.64; .164; (11)3.
4. What is the square of 13.68? of 9%?

5. What is the difference between 34 and 43? 6. What is the value of 117; 37; 23 × 22 ?

7. What power of 9 is equivalent to 95 × 93; 92× 910; 94 × 96; 9× 97 × 98 ?

8. Multiply 1279 by 1277, and divide the product by 12715.

9. Divide 319 by 319; 178 by 175; 427 by 426.

10. What is the sixth power of 40?

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11. What is the 9th power of 53? the 12th power of 185 the 24th power of 172?

CHAPTER XVI.

EVOLUTION.

exponent.

EVOLUTION is the process by which we discover the root of any given power. Thus, 3 is the 2d root of 9, the 3d root of 27, the 5th root of 243, because 9=32, 27=33, 243-35. So the 2d or square root of 49 is 7; the 3d or cube root of 125 is 5; the 4th root of 16 is 2; the 5th root of 1024 is 4, &c. We may denote a root by a radical sign, or by a fractional The radical sign is √ and when employed by itself denotes the square root. If we wish to denote the 3d, 5th, 7th, &c. root, the index of the root is written above the radical sign thus,,, &c. In fractional exponents, the numerator expresses the power of the number, and the denominator expresses the root.

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= 2/27; (16)2 = 4/163 ; (32)3 = ↓/ 324, &c.

Thus, (27)3

The product, or the quotient, of two second, third, or other roots, is the 2d, 3d, &c., root of the product or quotient. Thus, 3/27 × 3/125/27 × 125 or 3375. For 27=33=3×3× 3, and 125=5×5×5. Then 27 × 125= 3×3×3×5×5×5=3×5×3×5×3×5=153. Therefore 3/27×125=15. In a similar manner it may be shown

that /3375/125/27.

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The power of any root may be obtained by multiplying the fractional exponent. Thus the 4th power of 273= 27. For by the last proposition 3/272 × 3/272 × 3/272

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The root of any power or root may be obtained by dividing the exponent by the index of the desired root. Thus 3/33_315.

This is the converse of the last proposition. For if the 3d power of 315 is 315, or 35, the 3d root of 35 must be 31%.

If the numerator and denominator of fractional indices be multiplied or divided by the same number, the value of

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2

the quantity is not altered. Thus, 36-312-33. For the multiplication of the numerator involves the number to a certain power, and the multiplication of the denominator extracts the corresponding root. Then the 3d root of the 3d power, the 5th root of the 5th power, &c., is the 1st power.

5

=

We may multiply or divide any two roots of the same number, by adding or subtracting the fractional exponents. Thus, 2× √22+1=26; 5÷15—5—4—512. For by the last proposition we have 32× √2=22× 23, which is equivalent to 25 or 28. Also 35÷15

1

12/54 ÷ 12/53—12/5 or 512.

When the exact root of a number can be obtained, it is called a rational number. An irrational number, or surd, is one whose exact root cannot be obtained. Thus, 16,

27, 64, 81, are rational numbers, equivalent to 4, 3, 4, 3, respectively. But 5, 19, 16, are all surds, and their roots can only be obtained approximately.

A number which has a rational root, is called a perfect power. Thus, 16 is a perfect 2d power, and a perfect 4th power, but an imperfect power of any other degree. But 5, 7, 12, &c., are imperfect powers of any degree.

1. What is the square root of 9? the cube root of 8?

2. What is the 4th root of 81? the 5th root of 32?

3. What is the value of I; 25; 64; 6418? 4. What is the product of 48 by V12; 79 by 74? 5. Multiply 3 by 3; 72 by 2/7; 4% by √/42. 6. Divide 63 by 63; 4/5 by 5; 17 by 17.

7. Find the 4th power of √9; the 6th power of 8. What is the cube root of 76; the 5th root of 114?

EXTRACTION OF THE SQUARE ROOT.

The square root of a number is the number which, when multiplied by itself, will produce the given number.

In the following table are the numbers from 1 to 10 inclusive, and beneath them are their squares; therefore, the numbers of the second line have for their square roots the numbers of the first.

Roots 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

If there are decimals in the root, there will be twice as many in the square: because any product contains as many decimals as both factors. And conversely, there will be half as many decimals in the root as in the square.

Every entire number, which is not the square of another entire number, is an imperfect second power. For the root of such a number cannot be expressed by a fraction, because a fraction multiplied by itself would give a frac tional product, it must therefore be a surd.

The least square of units, is..
The least square of tens is.

The least square of hundreds, is.
The greatest square of units, is.

The greatest square of tens and units, is... . .
The greatest square of hundreds, tens and
units, is..

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Hence we may see that if we divide a square number into periods of two figures, by placing a point over units, and one over each second figure to the left, the number of periods will denote the number of figures in the root.

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