Second, let n➖➖r, a negative quantity, either integral or fractional, then I. Write the following expressions, using fractional exponents in place of the radical signs: 2. Write the following expressions, using radical signs in place of fractional exponents: x3, (x2+2xy+y}}, (x+y)3, 3. Multiply x † by x4. CHAPTER III. RADICAL QUANTITIES AND IRRATIONAL EXPRESSIONS. I. From the last chapter the student has learned that there are two methods in use for indicating the root of a quantity, one by the ordinary radical sign and the other by a fractional exponent. Of course it is entirely unnecessary to have two modes of writing the same thing, and in this sense either one of the two ways may be considered superfluous. But practically each method of notation has an advantage in special cases, and the student will feel this as he proceeds. This fact that both methods are better than either one, accounts for the retention of both in mathematics. 2. HISTORICAL NOTE-The introduction of the present symbols into algebra was very gradual, and the use of a particular symbol did not generally become common until some time after its suggestion. The signs and were first used at the beginning of the 16th century in the works of Grammateus, Rudolf and Stifel. Recarde (born about 1500) is said to have invented the sign of equality about this time. Scheubet's work (1552) is the first one containing the sign ✔✅, and Vieta (born 1540) first used the vinculum in connection with it. Before this, root-extraction was indicated by a symbol something like R. Stevin (born 1548) first used numbers to indicate powers of a quantity, and he even suggested the use of fractional exponents, but not until Descartes (born 1596) did exponents take the modern form of a superior figure. The development of the general notion of an exponent (negative, fractional, incommensurable) first appears in a work of John Wallis (published in 1665) in connection with the quadrature of plane curves. To show the appearance of mathematical works before the introduction of the common symbols, we give the following expression taken from Cardan's works (1545): Rv. cu. R 108 p. 10 m R cu. R 108 m 10, which is an abbreviation for "Radix universalis cubica radicis ex 108 plus 10, minus radice universali cubica radicis ex 108 minus 10." Or, in modern symbols, ✓108+10-108-10 Here is a sentence from Vieta's work (1615). Et omnibus per E cubum ductis et ex arte concinnatus, E cubi quad. + Z solido 2 in E cubum, acquabitur B plani cubo. This translated reads: Multiplying both members ("all") by E3 and uniting like terms, E 3 3. DEFINITIONS. In the following pages, by the word Radical may be understood the indicated root of an expression, whether that root is indicated by the ordinary radical sign or by a fractional exponent. By the Index of a radical may be understood either the number written in the angle of the radical sign or the denominator of the fractional exponent. A multiplier written before a radical will sometimes be called the co-efficient of the radical. A Simple radical is the indicated root of a rational expression. A Complex radical is the indicated root of an irrational expression. A monomial Surd is the name applied to the indicated root of a commensurable number, when that root cannot be exactly taken; as, or 3. If all the irrational terms in a binomial or polynomial are surds, it is called a binomial or polynomial surd, as the case may be. It should be noticed here that we make a distinction between the terms irrational expression and surd, a distinction which is not commonly made, the two terms being generally defined as identical. According to the above are not surds. But they are irraThis limited meaning of the word surd is It is found in both Aldis' and Chrystal's definition, 4, 2+√2, ✓ 3/3, √ R tional by the definition of I, Art. 3. convenient and is growing in use. algebras. Radicals are said to be Similar when they have the same index and the expressions under the radical signs are the same; that is, two radicals are similar when they differ only in their coefficients. Such are 5ab and m✔ab; also 3 and 4. 4. DEFINITION. For a radical to be in its simplest form it is necessary (1) that no factor of the expression under the radical sign is a perfect power of the required root; (2) that the expression under the radical sign is integral; (3) that the index of the radical is the smallest possible. It will be seen from the following pages that every simple radical can be placed in this form without changing its value. The transpositions necessary to effect the reductions depend upon certain principles, or theorems, established in the last chapter, which we collect here for reference. 5. The nth root of the product of several quantities is equal to the product of the nth roots of the several quantities. 6. The nth root of the quotient of two numbers is equal to the quotient of their n th roots. 7. The nrth root of a quantity equals the nth root of the rth root of the quantity. 8. TO REMOVE A FACTOR FROM BENEATH THE RADICAL SIGN. When any factor of the quantity beneath the radical sign is an exact power of the indicated root, the root of that factor may be taken and written as a coefficient while the other factors are left beneath the radical sign. Thus 128 may be written ✔64 × 2, which, by Art. 5, equals √64× √2, which equals 8√2. As another case take 16ax+, which equals No8x3× 2ax=8x3× ❖ 2ax =2x2ax. It is readily seen that this same process may be applied to any similar case. 9. EXAMPLES. Remove as many factors as possible from beneath the radical signs in the following: |