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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 a>, to show that a is repeated twice; and, instead of axaxaxa, we write a4. 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 a>; or, the 2d root of a”; 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.

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5

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The root of a quantity is expressed by means of the radical sign ✓ and an index. Thus, the 3d root of 8 is expressed ✓8, the 4th root of a, Va; the 5th root of x3, , &c.; a signifies that quantity whose 4th power is a; Væ% that quantity whose 5th power is x*, &c. Another mode of notation will be shown in the next chapter.

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)4; the 3d root of x8+ya+z8, x2 +y® +x, or ✓(22+ya+zo), &c.

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. Instead of ja we write simply Va.

5. Powers of the same quantity are multiplied together by adding their exponents. Thus, a xa’=aaaxaa=aaaaa=a”, and 5 is the sum of 3 and 2.

In general am xar=am+n; for am=aaa ..... repeated m times, and a"=aaa ....

repeated n times, so that am xan=aaa . repeated m+n times, which is the (m+n)th power of a, or am+n. In the same way, am xan X al Xar....=am+n+p+q.. for any number of factors.

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

a5 Thus, as divided by a>, is aaaaa divided by aa; that is,

a> =aaa=a', and 3 is the difference of 5 and 2.

aaaaa

aa

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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 a' and bs is abs; the

a? quotient of a divided by 63 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 a2b3c and a’df is

2oy6z6 a5b3cdf; the quotient of x5y6z7 divided by axz is and amonc

am-p fnac divided by apbed is

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8. Involution of powers is effected by multiplying the exponents.

Thus, to involve a’ to the 3d power, we are to take as three times as a factor, which is a xao xa”, or a2+2+3, or a®, (Art. 5,) that is, the 3d power of ao is a with the exponent 2x 3.

In general the nth power of am is amn; for (am)"=am xam xam repeated n times, which gives am+m+m..... with the exponent m repeated a times; that is, with the exponent mn. Hence (am)"=ann. So also (@mn)P=amne; (2mn)P=XnPa.

9. The evolution of powers is effected by dividing the exponents.

Thus, to extract the square root of at we divide the exponent by 2, which gives a. The 4th root of aểo is as ; the 5th root of y30 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 a b3

a: is a$613 ; the 5th power of 7 is ; the 4th root of 21%y* is x*y; the

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ma cube root of

monopo тп*р* is ; the cube root of is

2°y1% XPy4

015 Q10

mo n29

n3

11. Since (abc ....)m=ambmcm ....; and Wambacm .... 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.

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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 root of a is represented thus, Va; the 4th root of a?, Vas, &c. Hence, when it is required to extract the root of the product or quotient of several 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 aạb, we can take the square root of a”, which is a, but the square root of b we must denote by the radical sign, thus, vb; so that the square root of ałb is avb. In the same way we find the square root of a4b3 to be a b8. But this last expression may be reduced to a simpler form, thus ; a®/b3=ab2b, in which last form we can extract the square root b>, but must indicate that of b, so that the expression becomes aạbvb. In a similar manner we find,

4th root of a$611=7a*b*b8=ab% 96.
7th root of ab15=7ab146=01ab.
nth root of anym=Xym.

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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

32a107 is 2ņa; the square root of 16a8b is 4a7ab; the 5th root of

5 is 2a v

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Sometimes it is convenient to extract the root of the numeral part, even when it is not an exact power. Thus, V30abc=5.477a7bc 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)x(-a), which,

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