1. The square root of x. 3. The cube root of x2 + y.. 4. The fifth root of 79. 5. The square root of a2 b + 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. is the cube root of a. (a2 + b) is the fourth root of a2 + b. a is the cube root of a2. a3 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. 9. The m th root of c. 1 10. The square root of x3 c2 + 12. 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. 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 5 a3 is the square root of a3, multiplied by 5. or 5a3, implies that the square product of the But 5+ √ a3, root of a3 is to be added to, or subtracted from, 5, and not multiplied by that number. SECTION II. Roots of Simple Quantities 1. What is the square root of a ? ANS. a3. We are here required to find two equal factors, whose product shall be a6; and, as we multiply powers by adding their exponents, [See Chap. VII. Sec. VI.] a3 × a3 = a6. 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 a6? ANS. a2. Here we are required to find three equal factors, whose continued product shall be a6; 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 a. For 4 a X 4 a 16 a2. The root of the coeffi cient is found and prefixed to the root of the literal quantity, which is obtained as above. 4. What is the square root of 9 aa b2 x6 ? ANS. 3 a2 b x3. For 3 a2 bx3 × 3 a2 b x3 = 9 a4 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. q. 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 ca? ANS. 3 a ct. 7. Required the fifth root of 1024 a5 x10. ANS. 4 a x. 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 root 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 as x4 y12? 11. Required the fifth root of 32 x5 y10. 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 yo ? 15. Extract the fourth root of 1296 a4 b8 x16. 16. Required the fourth root of 16 as b3. 17. What is the third root of 9 a3 b4 x6 ? 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 manner. 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 9 a2 ? 16 b4 27 a6 8b6c3 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 — a = + a2, and + a2 × =- a3. And the cube root of `+ a ×+ a = + a2, and + a2 × α a3 is +a; for + a = + a3. An EVEN root of a positive quantity has two signs, the one positive, the other negative. Such a quantity is said to be ambiguous. The square root of a2 may be either +a, or for+a+a = + a2, and ах a; a = + a2, also. When it is not known, 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 negative quantity; for neither ах - a, nor+ax+a, 27. What is the square root of 25 a2 b1? 28. What is the cube root of 125 a3 b6? |