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BINOMIAL THEOREM

AND

LOGARITHMS.

CHAPTER I.

POWERS AND ROOTS.

1. The student is here supposed to be acquainted with the fundamental principles of algebra; at least, with simple equations, fractions, radicals and the multiplication, division &c. of powers; but, before proceeding to the consideration of the Binomial Theorem, we shall give, in the first and second chapters, a summary view of the nature of powers and roots, and of exponents in general.

2. The power of any quantity is that quantity multiplied by itself a certain number of times. Thus, axa is a power of a which is called the second power of a, because a is taken twice as a factor; and, uxaxaxa is the fourth power of a, because a is taken four times as a factor. But instead of repeating the letter in this manner, we indicate the operation by a small figure; thus, instead of axa, we write ao, to show that a is repeated twice; and, instead of axaxaxa, we write a1. This figure is called the exponent of the power.

3. The root of any quantity is a quantity which, multiplied by itself a certain number of times, produces the first quantity. Thus, 2 is the 3d root of 8, because 2x2x2=8. That is, 8 is the 3d power of 2, and 2 is the 3d root of 8. So also, a is the 3d root of a3; or, the 2d root of a3; or, the 4th root of a*, &c.

Power and root are correlative terms; am is the mth power of a, and a is the mth root of am.

The root of a quantity is expressed by means of the radical sign

3

✔ and an index. Thus, the 3d root of 8 is expressed 8, the 4th

4

5

root of a, a; the 5th root of 23, 3, &c.; a signifies that

5

quantity whose 4th power is a; √x that quantity whose 5th power is x2, &c. Another mode of notation will be shown in the next chapter.

3

The power or root of a compound quantity is expressed with the aid of the vinculum or parenthesis. Thus, the 4th power of a+b is expressed (a+b)*; the 3d root of x2+y3+z3, √x2+y2+z3, or √(x2+y3+z3), &c.

3

4. The second power is frequently called the square, the second root the square root; the third power, the cube, and the third root, the cube root.

The square root is denoted by the radical sign without the index.

2

Instead of a we write simply ✔a.

5. Powers of the same quantity are multiplied together by adding their exponents. Thus, a3×a2=aaa×aa=aaaaa=a5, and 5 is the sum of 3 and 2.

.....

In general am Xan=am+n; for am=aaa repeated m times, and anaaa .... repeated n times, so that am Xaa=aaa..... repeated m+n times, which is the (m+n)th power of a, or am+n. In the same way, a1×a1×аo ×аa....=am+n+p+q.... for any number of factors.

6. The division of powers of the same quantity is effected by subtracting their exponents.

Thus, as divided by a2, is aaaaa divided by aa; that is,

=aaa-a3, and 3 is the difference of 5 and 2.

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a5 ааааа

a3 αα

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repeated n times; so that

m―n times, which is the (m—n)th power of a, or am—n.

7. The multiplication and division of powers of different quanti

ties are merely represented. The product of a3 and b3 is ab3; the

a2

b3

quotient of a divided by b3 is Sometimes quantities to be multiplied or divided contain powers of the same quantity; these may be multiplied or divided by the above rules, and the operation upon the rest simply denoted. Thus, the product of a b3c and a3df is a5b3cdf; the quotient of xyz7 divided by axz is

α

and ambac

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8. Involution of powers is effected by multiplying the exponents. Thus, to involve a2 to the 3d power, we are to take a three times as a factor, which is a xaxa, or a2+2+9, or a®, (Art. 5,) that is, the 3d power of a2 is a with the exponent 2 × 3.

.....

In general the nth power of am is amn; for (am)n=am×am xam repeated n times, which gives am+m+m..... with the exponent m repeated a times; that is, with the exponent mn. Hence (am)n=amn. So also (am)=ɑmn; (xmn)=xmnpq.

n

9. The evolution of powers is effected by dividing the exponents. Thus, to extract the square root of a we divide the exponent by 2, which gives a3. The 4th root of a20 is a5; the 5th root of y3° is yo, &c. The proof follows from what was shown in the last article, since evolution is the inverse of involution. In general the nth root

mn

of amn is an or am.

10. The involution and evolution of the product or quotient of powers are effected in the same manner. The 4th power of a2b3

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m

11. Since (abc....)m=ambmcm....; and ambmcm.... abc...., we have the following principles :

The power of the product of any number of quantities is equal to the product of their powers; and, the root of the product of any number of quantities is equal to the product of their roots.

12. The root of a quantity which is not an exact power (or if a power, one whose exponent is not a multiple of the index of the root) can only be represented. For example, the 5th

5

4

sented thus, ✔a; the 4th root of a3, a3, &c.

root of a is repreHence, when it is

quotient of several

required to extract the root of the product or quantities, it frequently happens that we can perform the operation in respect to a part of the quantities only, and must merely represent it in the case of the others. Thus, to extract the square root of ab, we can take the square root of a3, which is a, but the square root of b we must denote by the radical sign, thus, ✅b; so that the square root of a'b is ab. In the same way we find the square root of a1b3 to be a3✓b3. But this last expression may be reduced to a simpler form, thus; ab3=abb, in which last form we can extract the square root b3, but must indicate that of b, so that the expression becomes ab√b. In a similar manner we find, 4th root of asb11=a*b*b3⁄4=a2b2 Ŷb3.

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13. When the quantities have numeral coefficients, the operations of involution and evolution may always be performed upon these, and the result prefixed to the literal part. Thus, the 3d root of 8a 32a10b is 2ỷa; the square root of 16a3b is 4a√✅ab; the 5th root of

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C

Sometimes it is convenient to extract the root of the numeral part, even when it is not an exact power. Thus, ✔30a2bc=5.477abc nearly.

14. Thus far we have said nothing of the signs of powers and

roots.

If we involve —a to the 2d power, we have (—a)×(—a), which,

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