« ΠροηγούμενηΣυνέχεια »
The second power is commonly called the square; the third power, the cube; the fourth power, the biquadrate. The other powers now generally receive no other than numeral distinctions; as the fifth power, the sixth power, the seventh power, &c. In some books, however, the fifth power is called the first sursolid; the sixth power, the square cubed, or the cube squared; the seventh power, the second sursolid; the eighth power, the biquadrate squared; the ninth power, the cube cubed.
The powers of 1 remain always the same; because, whatever number of times we multiply 1 by itself, the product is always 1.
A power is sometimes denoted by a number, placed at the right hand of the upper part of the root; thus, 52 denotes the second power of 5, which is 25; 43 denotes the third power of 4, which is 64; 94 denotes the fourth power of 9, which 6561; &c. The number, thus used to denote the power, is sometimes called the exponent and sometimes the index. But the use of these exponents or indices in arithmetic is very limited; they belong chiefly to algebra.
We will now make a few observations on the result arising from the multiplication or division of one power by another. To illustrate this subject, we will take the number 3; we must here observe, however, that since every number is the first power of itself, the exponent 1 is never expressed; so that 3 and 3? mean the same thing; the exponent 1 being always understood, when no exponent is expressed. Now 3 multiplied by 3 produces the second power of 3, which may be thus expressed, 31 x 31=32; so also 32 X 31=33; and 33 x 31=34, &c. We have here expressed the exponent 1 for the purpose of showing that we obtain the exponent of the product or power produced, by adding together the exponents of the factors or powers used in producing it. Hence the second power of any number multiplied by the second power of the same number produces the fourth power of that number; thus, 32 X 32=34: the third power multiplied by the third power gives the sixth power; as 23 x 23=26: the fourth power multiplied by the second power gives the sixth power; as 24 X 22 =26: the fourth power multiplied by the fourth power produces the eighth power; as 34 x 34=38: the third power multiplied by the third power, and the product again by the third power gives the ninth power; as 23 x 23 x 23=2?.
Division being the reverse of multiplication, it is evident, that if we subtract the exponent of the divisor from the exponent of the dividend, the remainder is the exponent of the quotient. For example, if we divide the fifth power by the third power, the quotient is the second power; as 35:33=32: if we divide the ninth power by the sixth power, the quotient is the third power; as 6° = 66=63: if we divide the ninth power by the eighth power, the quotient is the first power; as 69=6856.
1. What is the third power of 12?
8 Roots. 1 2 3 4 5 1 6 bal
8th. Pow. 1| 256 6561 65536 390625 / 1679616 5764801 | 16777216 430467218
Evolution is the reverse of involution; for in involution we have the root given, to find the power; but in evolution we have the power given, to find the root.
Power and root are correlative terms; for, as 4 is the square of 2, 2 is the square root of 4; as 8 is the cube of 2, 2 is the cube root of 8; as 16 is the biquadrate of 2, 2 is the biquadrate root of 16; as 32 is the fifth power of 2, 2 is the fifth root of 32: &c.
The extraction of the root is finding a number, which being multiplied into itself the requisite number of times, will reproduce the given number: for example, if we extract the square root of 81, we find it to be 9, because 9 X 9=81; but if we extract the biquadrate root of 81, we find it to be 3, because 3 X3 X3 X3=81.
Hence the root is designated by the number of times it is used as factor in producing the corresponding power. It is used twice in producing the second power, and is called the second root, or square root: it is used three times in producing the third power, and is called the third root, or cube root: it is used four times in producing the fourth power, and is called the fourth root, or biquadrate root: it is used five times in producing the fifth power, and is called the fifth root: &c. : *
A number, whose root can be exactly extracted, is called a perfect power, and its root is called a rational number. For example, 4 is a perfect power of the second degree, and 2, its square root, is a rational number; 27 is a perfect power of the third degree, and 3, its cube root, is a rational nunber; 64 is a perfect power of the second, third, and sixth degrees, and 8 its square root, 4 its cube root; and 2 its sixth root, are rational numbers; , is a perfect power of the third degree, and 3, its cube root, is a rational number; .25 is a perfect power of the second degree, and .5, its square root, is a rational number.
In short, any number, which is the exact root of any power, is a rational nuniber, and its power a perfect power: and since any number may be the root of its corresponding power, it follows that any root, which can be exactly expressed by figures, is a rational number.
But there are numbers, whose roots can never be exactly extracted, and these numbers are called imperfect powers, and their roots are called irrational numbers, or surds. For example, 2 is not only an imperfect power of the second degree, but an imperfect power of any degree, and not only its square root, but the root in every degree is irrational, or a surd; because no number, either whole or fractional, can be found, which, being involved to any degree, will produce 2. The same is true of many other numbers.' In these cases, by using decimals, we can approximate, or come very near to the root, which is sufficient for most purposes. Thus, we find the square root of 2 to be 1.414+. The decimal may be carried to any number of places.
Some numbers are perfect powers of one degree, and imperfect powers of another degree. For example, 4 is a perfect power of the second degree, and its square root, which is 2, iş rational; but an imperfect power of the third degree, and its cube root, which is 1.587 +, is a surd: 8 is an imperfect power of the second degree, and its square root, which is 2.928 t, is a surd; but a perfect power of the third degree, and its cube root, which is 2, is rational: 16 is a perfect power of the second and fourth degrees, and its square root, which is 4, and its biquadrate root, which is 2, are both rational; but an imperfect power of the third degree, and its cube root, which is 2.519 +, is a surd.
These irrational numbers or surds occur, whenever we endeavor to find a root of any number, which is not a perfect corresponding power; and, although they cannot be expressed by numbers either whole or fractional, they are nevertheless magnitudes, of which we may form an accurate idea. For however concealed the square root of 2, for example, may appear, we know, that it must be a number, which, when multiplied by itself, will exactly produce 2. This property is sufficient to give us an idea of the number, and we can approximate it continually by the aid of decimals.