CHAPTER VII.

EXTRACTION OF ROOTS, AND RADICALS.

EXTRACTION OF ROOTS.

128. Evolution is the process of finding the equal factors of a number or quantity. One of such factors is called a root of the quantity, and the operation of finding it is called extracting the root. The sign is called the radical sign. A number placed above and in the opening of the radical sign indicates the root to be extracted, and is called the index of the root. Thus, in 81, 3 is the index of the root, and shows that the cube root is to be found; in √32a1, 5 is the index of the root. When the index is 2, it may be omitted.

Instead of the radical sign and index to denote a root, a fractional exponent is often used, in which the index is the denominator. Thus, instead of Va, we may use

=

a

to denote

the cube root of a; also Vaa3, √bb, etc. The square root of a number is one of its two equal factors. Thus, 6 x 6 36 therefore 6 is the square root of 36. The symbol for the square root is or the fractional exponent. Thus,

=

√a, or a3,

indicates the square root of a, or that one of the two equal factors of a is to be found.

Square Root.

129. A perfect square is any number which can be resolved into two equal integral factors.

The following table, verified by actual multiplication, indicates all the perfect squares between 1 and 100:

Squares.
Roots

1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

We may employ this table for finding the square root of any perfect square between 1 and 100.

Look for the number in the first line. If it is found there, its square root will be found immediately under it.

If the given number is less than 100, and not a perfect square, it will fall between two numbers of the upper line, and its square root will be found between the two numbers directly below. The lesser of the two will be the entire part of the root, and will be the true root to within less than 1.

Thus, if the given number is 55, it is found between the perfect squares 49 and 64, and its root is 7 and a decimal fraction.

NOTE. There are ten perfect squares between 1 and 100, if we include both numbers; and eight, if we exclude both.

If a number is greater than 100, its square root will be greater than 10; that is, it will contain tens and units. Let N denote such a number, x the tens of its square root, and the units. Then will

y

N = (x + y)2 = x2 + 2xy + y2 = x2+(2x+y) y ; that is, the number is equal to the square of the tens in its roots, plus twice the product of the tens by the units, plus the square of the units.

60 84

For example, extract the square root of 6084. Since this number is composed of more than two places of figures, its root will contain more than one but since it is less than 10,000, which is the square of 100, the root will contain but two figures;

that is, units and tens.

Now, the square of the tens must be found in the two lefthand figures, which we will separate from the other two by putting a point over the place of units, and a second over the place of hundreds. These parts, of two figures each, are called periods. The part 60 is comprised between the two squares 49 and 64, of which the roots are 7 and 8: hence 7 expresses the number of tens sought; and the required root is composed of 7 tens and a certain number of units.

The figure 7 being found, we write it on the right of the given number, from which we separate it by a vertical line: then we subtract its square, 49, from 60, which leaves a remainder of 11, to which we bring down the two

60 84178 49

7x2=148 118 4

118 4

next figures, 84. The result of this operation, 1184, contains twice the product of the tens by the units, plus the square of

the units.

But since tens multiplied by units cannot give a product of a less unit than tens, it follows that the last figure, 4, can form no part of the double product of the tens by the units. This double product is therefore found in the part 118, which we separate from the units' place, 4.

Now, if we double the tens, which gives 14, and then divide 118 by 14, the quotient 8 will express the units, or a number greater than the units. This quotient can never be too small, since the part 118 will be at least equal to twice the product of the tens by the units; but it may be too large, for the 118, besides the double product of the tens by the units, may likewise contain tens arising from the square, of the units. To ascertain if the quotient 8 expresses the right number of units, we write the 8 on the right of the 14, which gives 148, and then we multiply 148 by 8. This multiplication, being effected, gives for a product 1184, a number equal

to the result of the first operation. Having subtracted the product, we find the remainder equal to 0: hence 78 is the root required. In this operation we form, first, the square of the tens; second, the double product of the tens by the units; and, third, the square of the units.

Indeed, in the operations, we have merely subtracted from the given number 6084: first, the square of 7 tens, or of 70; second, twice the product of 70 by 8; and, third, the square of 8, that is, the three parts which enter into the composition of the square, 70+8, or 78; and since the result of the subtraction is 0, it follows that 78 is the square root of 6084.

130. The operations in the last example have been performed on but two periods; but it is plain that the same methods of reasoning are equally applicable to larger numbers, for, by changing the order of the units, we do not change the relation in which they stand to each other.

Thus, in the number 60 84 95, the two periods 60 84 have the same relation to each other as in the number 60 84; and hence the methods used in the last example are equally applicable to larger numbers.

131. Hence, for the extraction of the square root of numbers, we have the following rule:

Point off the given number into periods of two figures each, beginning at the right hand.

Note the greatest perfect square in the first period on the left, and place its root on the right, after the manner of a quotient in division; then subtract the square of this root from the first period, and bring down the second period for a remainder.

Double the root already found, and place the result on the left for a divisor. Seek how many times the divisor is contained in the remainder, exclusive of the right-hand figure, and place the figure in the root and also at the right of the divisor.

Multiply the divisor thus augmented by the last figure of the root, and subtract the product from the remainder, and bring down the next period for a new remainder. But if any of the products should be greater than the remainder, diminish the last figure of the root by one.

Double the whole root already found, for a new divisor, and continue the operation as before until all the periods are brought down.

132. If, after all the periods are brought down, there is no remainder, the given number is a perfect square.

The number of places of figures in the root will always be equal to the number of periods into which the given number is separated.

If the given number has not an exact root, there will be a remainder after all the periods are brought down; in which case ciphers may be annexed, forming new periods, for each of which there will be one decimal place in the root.