If the numerator and denominator of a fractional index be the same, the value of the quantity is not affected by it; for a3, 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, a, a, a, &c., which are all equal.

2

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, a implies that the square root of x is multiplied by a; and 5a3 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.

SECTION II.

Roots of Simple Quantities

1. What is the square root of ao ?

ANS. a3. We are here required to find two equal factors, whose product shall be a; and, as we multiply powers by adding their exponents, [See Chap. VII. Sec. VI.] a3 × a3 = a. Or the required root may be expressed by a fractional index, thus, a&; which, the fraction being reduced, becomes a3.

2. What is the cube root of a?

ANS. a2.

Here we are required to find three equal factors, whose continued product shall be a; and, by the rule for multiplying powers, a2 × ̧a2 × a2 = ao. If the required root be expressed by a fractional index, it will be a a2, as above.

3. What is the square root of 16 a2?

ANS. 4 α.

For 4 a X 4 a 16 a2. The root of the coefficient is found and prefixed to the root of the literal quantity, which is obtained as above.

4. What is the square root of 9 a1 b2 x6 ?

ANS. 3 a2 b x3.

For 3 a2 b x3 × 3 a2 b x3 = 9 aa b2 x6. We divide the exponent of every letter by the index of the required root, and annex the result to the root of the coefficient.

5. What is the cube root of a5?

ANS. a

As the exponent of the given power cannot be divided by the index of the required root, without leav

ing a remainder, the root must be represented by a fractional index.

6. What is the fourth root of 81 a4 c3?

7. Required the fifth root of 1024 a5 x10.

ANS. 4 a x2.

From these examples and observations we derive the following RULE for extracting the root of a simple quantity, viz:

Divide the exponent of the given power by the index of the root to be found, and annex the result to the rout of the coefficient.

8. What is the square root of 64 aa b2?

9. What is the cube root of 27 a3 b6 x9 ? 10. What is the fourth root of 81 a8 x4 y12? 11. Required the fifth root of 32 x5 y1o. 12. Extract the cube root of 64 a6 x3 y12. 13. Required the square root of 5 a2 x4. 14. What is the cube root of 7 x6 y3 ? 15. Extract the fourth root of 1296 a1 b8 x16. 16. Required the fourth root of 16 a8 63. 17. What is the third root of 9 a3 b4 x¤?

18. Find the square root of 25 aa b.

19. Extract the cube root of 64 xa y5.

The root of a fraction is found in the same man ner. Extract the root of the numerator for a new numerator, and the root of the denominator for a new denominator.

20. What is the square root of

21. Required the cube root of

9a2

16 b4

27 a6 866 c3

ܕ܂

To determine what sign should be prefixed to a root, observe, in general, that the root, when multiplied by itself the requisite number of times, must re-produce the given power. Therefore,

An ODD root of any quantity must have the same sign which the quantity has. The cube root of a3

is

αχ a; for

=

α

— a + a2, and + a2 × a3. And the cube root of a3 is+a; for + a×+ a = + a2, and + a2 × + a = + a3. An EVEN root of a positive quantity has two signs, the one positive, the other negative. Such a quantity es said to be ambiguous. The square root of a2 may be either+a, or a; for a x + α= + a2, When it is not known,

αχ

a = + a2, also.

and

from the nature of the question, whether the root is positive or negative, it should be marked with the ambiguous sign; thus ±a.

There is no such thing as the EVEN root of a nega tive quantity; for neither αχ a, nor+ax+α,

a2.

27. What is the square root of 25 a2 b1? 28. What is the cube root of 125 a3 b6 ?

29. What is the square root of 7 x4 y3?

30. Required the fifth root of 243 x1o. 31. Extract the fourth root of 256 as b4 c12. 32. Required the square root of 64 a2 xa. 33. Extract the cube root of 125 mo n3.

-

34. What is the sixth root of 4096 x6 z12? 35. Required the cube root of 64 x3 z6. 36. Extract the square root of

37. What is the cube root of

38. Required the fifth root of

39. Find the cube root of

17 x4 y3 22.

18 α6 y1 z3?

a10 b5

18c5 x

12 a3 x7

125 y 24

SECTION III.

To extract the Square Root of a Compound Quantity.

1. What is the square root of 12 x2 y4 + 4 yε + 9x4 y2?

Since the power given in this question consists of three terms, it is evident that its root must contain more than one term; for the second power of a simple quantity is a simple quantity, and the second power of a binomial consists of three terms. [See Chap. VII. Sec. III.] We are, therefore, required to find the binomial, whose square is the quantity proposed in the question.

We may here remark that no binomial, as x2 + y2, can be a complete power; and the root of an incomplete power can be found only by approximation.