SECTION XXVIII.

SECOND ROOTS OF FRACTIONS, AND EXTRACTION OF SECOND ROOTS BY APPROXIMATION.

ART. 87. A fraction is raised to the second power by raising both numerator and denominator to that power, this being equivalent to multiplying the fraction by itself.

Consequently, the second root of a fraction is found by extracting the root of both numerator and denominator.

ART. 88. But if either the numerator or denominator is not an exact second power, we can find only an approximate root of the fraction. Thus, the root of 15 is between and, or 1, the latter being nearer the true root than the former.

We can always make the denominator of a fraction a perfect second power, by multiplying both numerator and denominator by the denominator. This does not change the value, but only the form of the fraction. For example,

Remark. The sign +, placed after an approximate

root, denotes that it is less, and the sign greater, than the true root.

that it is

If a greater degree of accuracy is required, we may, after preparing the fraction as above, multiply both numerator and denominator of the result by any second power, and then extract the root. Thus, to find the root

of ; after changing it to §, we may multiply the numerator and denominator of § by 225, the second power the approximate root of which

of 15; this gives

14175 81.225'

ART. 89. The roots of whole numbers, which are not exact second powers, may be approximated in a similar manner. For example, to find the root of 3, accurate to within, we convert it into a fraction having the second power of 15 for its denominator. Thus, 3=, the root of which is 2

But the most convenient number for a denominator is the second power of 10, 100, or 1000, &c.; that is, we convert the number into 100ths, 10000ths, or 1000000ths, &c., and the root will be in decimals.

00000

Thus, 3388=18888=1888888; that is, 33.00 =3·0000=3·000000. The approximate root of the first 18+7+=1·7+; that of the second is 178+=1·73+; that of the third, 1733 += 1·732 +.

It is evident that there must be twice as many decimals in the power as we wish to find in the root; for the second power of 10ths produces 100ths, the second power of 100ths produces 10000ths, &c. Hence two zeros must be annexed to the number for each additional decimal in the root. Nor need the zeros be all written at once, but we may annex two zeros to each remainder, in the same manner as we bring down successive periods.

Let us, in this way, extract the second root of 5.

Operation.

5(2.236. Root.

4

10'0 (42

84

160/0 (443

1329

2710'0 (4466

26796

304.

The operation may be continued to any desirable

extent.

When the given number contains decimals, the process of finding the root is the same; and any fraction may be changed to decimals and the root be found in the same way, care being taken, in both cases, to make the number of decimals even, and to point off half as many figures for decimals in the root, as there are in the power, including the zeros annexed.

In separating a number containing decimals into periods, it is best to begin at the decimal point, and separate the decimals by proceeding towards the right, and the integral numbers by proceeding towards the left.

Let the roots of the following numbers be found in decimals, each root containing three decimal figures.

SECTION XXIX.

QUESTIONS PRODUCING PURE EQUATIONS OF THE SECOND

DEGREE

ART. 90. 1. Two numbers are to each other as 2 to 3, and their product is 96. Required the numbers

2. What number is that whose part produces 48?

part mulupied by its

3. The length of a house is to its breadth as 10 to 9, and it covers 1440 square feet of land. Required the length and breadth.

4. What number is that to which if 5 be added, and from which if 5 be subtracted, the product of the sum and difference shall be 39?

5. A man bought a farm, giving as many dollars per acre as there were acres in the farm, and the whole amounted to $2000. Required the number of acres and

the price per acre.

6. Two numbers are to each other as 8 to 5, and the difference of their second powers is 156. Required these numbers.

7. A gentleman has a rectangular piece of land 25 rods long and 9 rods wide, which he exchanges for a square piece of the same area. Required the length of one side of the square.

8. The sides of two square floors are to each other as 7 to 8, and it requires 15 square yards more of carpeting to cover the larger, than it does to cover the smaller. Required the length of one side of each floor.

9. A farmer bought two equal pieces of land, giving for the whole $1800. For one he gave $10 less, and for

the other $10 more, per acre, than there were acres in each piece. Required the number of acres in each. 10. The product of two numbers is 500, and the greater divided by the less gives 5. What are the num

bers?

11. An acre contains 160 square rods. Required the length of one side of a square containing an acre.

12. What is the length of a square piece of land in which there are 5 acres?

13. A cistern having a square bottom, and being 4 feet deep, contains 600 gallons. Required the length of one side of the bottom, a gallon wine measure being 231 cubic inches.

SECTION XXX.

AFFECTED EQUATIONS OF THE SECOND DEGREE.

ART. 91. Equations of the second degree which we have hitherto considered, contained the second power, but no other power, of the unknown quantity. But an equation of the second degree, in its most general sense, contains three kinds of terms, viz., one in which there are two unknown factors; another in which there is but one unknown factor; and a third composed wholly of known quantities.

Equations of this description, when they contain only one unknown quantity, are called affected equations of the second degree, or affected quadratic equations.

1. The length of a rectangular piece of land exceeds its breadth by 6 rods, and the piece contains 112 square rods. Required the dimensions.