money 3 years; the interest being reckoned at 5 per cent. What was the sum lent?

79. In the course of 4 years, a man paid interest to the amount of $288, which was reckoned at 6 per cent. What was the debt?

80. The amount, principal and rate being given, to find the time.

That is, From the amount subtract the principal; and divide the remainder by the product of the rate and principal. The quotient will be the time.

81. A man lent $460 at 5 per cent. and received, for principal and interest, $529.

money kept?

How long was the

82. In what time will $780 amount to $1014, interest being reckoned at 6 per cent.?

These examples show the manner in which general results, or formulas, are obtained; and also how they may be used in solving particular questions. Let the learner now turn back to Chapter VIII., and generalize the questions marked with a star (*); and then solve the same questions, numerically, by their respective formulas. He will thus be prepared to generalize some of the more difficult questions, in the same chapter, which are not marked.

CHAPTER X.

EVOLUTION.

SECTION I.

Introduction.

WHEN a quantity is multiplied by itself one or more times, the product is called a Power of that quantity. Thus, a2, being the product of a Xa, is the secon❜l power or square of a; and b3, that is, b xbx 1, is the third power or cube of b. [See Chap. VII. Sec. I.]

On the contrary, the quantity which is multiplied by itself to produce any power, is said to be the Root of that power. Thus, a is the second or square root of a2; and b is the third or cube root of b3.

Powers and Roots are, therefore, correlative terms; and Evolution and Involution are the reverse of each other. Involution is the method of raising a given root to a proposed power; but Evolution is the method of finding the roots of given powers.

Involution is more perfect, however, than Evolution; for if any proposed power of a given quantity be required, it can be exactly obtained; but there are many quantities whose exact roots cannot be found.

It is evident, for instance, that the square root of a cannot be determined; for there is no quantity, which, being multiplied by itself, will produce a.

The roots and powers of numbers have the same relation to each other as those of literal quantities. Thus, the second powers of 2 and 3 are 4 and 9; and the square roots of 4 and 9 are 2 and 3. The exact roots of the intermediate numbers, 5, 6, 7 and 8, cannot be found.

TABLE OF ROOTS AND POWERS.

4th powers. 1 16 81 256 625 1296 2401 4096 6561 10000 5th powers. 1 32 243 1024 3125 7776 16807 32768 59049 | 10000)

The roots of quantities are indicated either by means of the radical sign √, or by a fractional index. Thus, 2a, or a, is the square root of a.

3

Va is the cube root of a.

a3 is the 4th root of a3.

64 is the square root of 64, which is 8.

3a +x is the cube root of a + x.

If the quantity affected by the radical sign be not a complete power, that is, if its root cannot be exactly found, it is called a Surd, or Irrational Quantity. Thus, 35, 3x2, 5a3, &c., are surd quantities. Express the roots of the following quantities by means of the radical sign:

1. The square root of x.
2. The fourth root of b3

3. The cube root of x2 + y.

4. The fifth root of 79.

5. The square root of a2b+14.

When the root of a quantity is expressed by means of a fractional index, the numerator of the fraction indicates the power of the quantity, and the denominator the root required.

Thus, a is the square root of a1 or a.

a is the square root of a3.

at is the cube root of a.

(a+b) is the fourth root of a2 + b.

a is the cube root of a2.

The expression a may be regarded either as the second power of the third root of a, or as the third root of the second power of a. And so with all other quantities having fractional indices.

Suppose the value of a to be 27. The third root of 27 is 3, and the second power of 3 is 9. Again, the second power of 27 is 729, and the third root of 729 is 9. The value is the same, whichever mode of expression is used.

Express the roots of the following quantities by means of fractional indices:

6. The square root of x.

7. The fourth root of y3.

8. The cube root of (a2 + x)2.

If the numerator and denominator of a fractional index be the same, the value of the quantity is not affected by it; for a2, that is, the second root of the second power of a, is evidently a.

As the value of a fraction is not altered, when both the numerator and denominator are either multiplied or divided by the same number, fractional indices may be changed into other indices of the same value; as, a2, aa, að, at, &c., which are all equal.

Suppose the value of a to be 16. Then the second root of a is 4, whose first power is also 4. Again, the fourth root of a, or 16, is 2; and the second power of 2 is 4. And so with the others.

We can, therefore, reduce different fractional indices to other indices which shall express the same root, by reducing the fractions to a common denominator.

When a letter or figure is prefixed to a quantity affected by the radical sign, it is to be regarded as a coefficient, and the two quantities are supposed to be multiplied together.

Thus, ax implies that the square root of x is multiplied by a; and 5 a3 is the product of the square root of a3, multiplied by 5. But 5+ a3, a3, implies that the square root of a3 is to be added to, or subtracted from, 5, and not multiplied by that number.

or 5