146. It follows from the principles established above, that if we multiply at the same time the index of the radical and the exponents of the quantity placed under the radical sign by the same number, the value of the radical remains the same. Thus if we multiply the index of the radical Vab by 3, 15 we have a b, the third root of the proposed; if then we multiply the exponent of the quantity placed under the radi cal sign by 3, we have ✔ a b3 the third power of a2 b, the second operation therefore restores the expression to its original value. 147. By means of this last principle, we may reduce two or more radicals of different indices to the same index. Thus let there be the two radicals ✔ 2 a, √b2 c. c. Multiplying the index and also the exponents of the quantities placed under the radical sign in the first by 4, and in the second by 3, we have for the first 2* a1 or 16 a1, and for the second✅bc3. The proposed are therefore reduced to equivalent expressions having a common index 12. In like manner the three quantities From what has been done we have the following rule for reducing radical expressions to the same index, viz. Multiply at the same time the index belonging to each radical sign, and the exponents of the quantities placed under this sign by the product of the indices belonging to all the other radical signs. If the indices of the radicals have common factors, the calculations are rendered more simple by taking for the common index the least number exactly divisible by each of the indi ces. A quantity, which has no radical sign, may on the same principles be placed under a radical sign; for this purpose, we raise the quantity proposed to the power denoted by the index of the radical sign, under which it is to be placed. شد Thus if it be required to put the quantity a2 under the sign 148. Radical expressions having different indices must be reduced to the same index before applying to them the rules for multiplication and division laid down above. The following examples will serve as an additional exercise in the multiplication and division of radical quantities. 4 4 4 8a by 2b4ac Ans. 12 ab Ŷ 2 c. 3 4. Multiply 3 a ✔ 3, and 6. Multiply ✔ 5. Multiply 2, a2 7. Multiply 26 −3/5 by 4 / 3 — √10. 42 Ans. 27. Ans. 39/2-16 √15. 8. Multiply 4 } + 5 √ 1⁄2 by √ } + 2 √ 1⁄2. 18. Divide ✔✔a+✔b by √ √ a−√b n Ans. 2 b a2 + b2 a2 Ans. V a+b+2ab .b α SECTION XIX. THEORY OF EXPONENTS. 149. We have seen, art. 42, that with respect to the same letter, division is performed by subtracting the exponent of the divisor from that of the dividend. The application of this rule to the case, in which the exponent of the divisor is equal to that of the dividend, gives rise to the exponent 0. An expression a°, in which this exponent is found, is to be regarded, art. 45, as a symbol equivalent to unity. 150. The application of the same rule to the case, in which the exponent of the divisor exceeds that of the dividend, gives rise to negative exponents. Thus let it be required to divide a3 by a3. Subtracting the exponent of the latter from that of the former, we have a 2 for the result. But a3 divid a3 ed by a3 is expressed by the fraction; reducing this fraction 1 to its lowest terms, we have The expression a 2 must therefore be regarded as equivalent to am In like manner+n gives by subtracting the exponent of the divisor from that of the dividend a"; but the fraction The expression a" is therefore the symbol of a division, which cannot be performed. Its true value is the quotient of unity divided by a raised to a power denoted by the negative exponent n. 151. To find the roots of monomials, we divide, art. 125, the exponents of the proposed by the index of the root required. The application of this rule to the case, in which the exponents of the proposed are not divisible by the index of the root, gives rise to fractional exponents. Thus let the third root of a be required. Indicating upon the exponent of a the operation required in order to obtain the third root, we have for the result a3. But we have agreed to indicate the 3 3 डै third root by; the expressionsa, a are therefore to be regarded as equivalent. In like manner, we have The expression a" is therefore to be regarded as a symbol equivalent to the nth root of the mth power of a. 152. The two preceding cases sometimes meet in the same expression. This gives rise to negative fractional exponents. Thus let it be required to extract the seventh root of a3 divid a3 a5 ed by a5; we have =a-2, the seventh root of which is a-4. In like manner the nth root of The expression a " is therefore the symbol of a division which cannot be performed, combined with the extraction of Its true value is the nth root of the quotient of unity di a root. vided by a raised to the mth power. above explained from rules previously established, have become by agreement notations equivalent respectively to 1, 1 1 am, Nam, Vami we may therefore at pleasure substitute the former of these expressions for the latter and the converse. 153. We proceed to show, that the rules already established for performing the operations of arithmetic upon quantities affected with entire and positive exponents are sufficient for these operations, whatever the exponents may be, with which the quantities are affected. MULTIPLICATION. 용 Let it be required to multiply a by a. To perform the operation required, it is sufficient to add the exponents. But adding the exponents of the proposed, we have m p Let it be required next to multiply a ̄ by ai |