exponent of each letter must be divisible by the index of the root. When the quantity whose root is required is not a perfect power of the given degree, we can only indicate the operation. to be performed. Thus, if it be required to extract the cube root of 4a2b3, the operation may be indicated by writing the expression thus, /4a2b°. Expressions of this nature are called surds, or irrational quantities, or radicals of the second, third, or nth degree, according to the index of the root required. (152.) The method of extracting the roots of polynomials will be considered in Section XVII. There is, however, one class so simple and of so frequent occurrence that it may properly be introduced here. In Arts. 60 and 61 we have seen that the square of and the square of a+b is a2+2ab+b2, a-b is a2-2ab+b2. Therefore, the square root of a±2ab+b' is a±b. Hence a trinomial is a perfect square when two of its terms are squares, and the third is the double product of the roots of these squares. Whenever, therefore, we meet with a quantity of this description, we may know that its square root is a binomial; and the root may be found by extracting the roots of the two terms which are complete squares, and connecting them by the sign of the other term. Ex. 1. Find the square root of a2+4ab+4b2. The two terms, a2 and 4b2 are complete squares, and the third term 4ab is twice the product of the roots a and 2b; hence a+2b is the root required. Ex. 2. Find the square root of 9a2-24ab+16b3. (153.) No binomial can be a perfect square. For the square of a monomial is a monomial; and the square of a binomial consists of three distinct terms, which do not admit of being reduced with each other. Thus such an expression as a2+b2 is not a square; it wants the term ±2ab to render it the square of a b. This remark should be continually borne in mind as beginners often put the square root of a2+b2 equal to a+b.' IRRATIONAL QUANTITIES, OR SURDS. (154.) A rational quantity is one which can be expressed in finite terms, and without any radical sign; as a, 5a2, &c. Irrational quantities, or surds, are quantities affected with a radical sign, and which have no exact root, or a root which can be exactly expressed in numbers. Thus, 3 is a surd, because the square root of 3 can not be expressed in numbers with perfect exactness. In decimals it is 1.7320508 nearly. (155.) We have seen, Art. 144, that in order to extract the square root of a monomial, we must divide each of its exponents by 2. Thus the square root of a' is a' or a; that of a' is a'; that of a' is a3, and so on; and as this principle is general, the square root of a' must necessarily be a3, and that of a' must be a3; 3 and, in the same manner, we shall have a for the square root of a'. Whence we see that We have also seen, Art. 149, that in order to extract any root of a monomial, we must divide the exponent of each letter by the index of the required root. Thus, the cube root of a3 is a', or a; the cube root of a® is a'; the cube root of a' is a3, and so on. So, also, the cube root of a2 is a31; the cube root of a is a; the cube root of a or a', is a3. Whence it appears that a3 is the same as Va, a3 is equivalent to Va, In the same manner, the fourth root of a is a3, which expres. sion has therefore the same value as Va; the fifth root of a will be a3, which is, consequently, equivalent to Va, and the same principle may be extended to all roots of a higher degree. (156.) Other fractional exponents are to be understood in he same way. Thus, if we have a3, this means that we must first take the fifth power of a, and then extract its fourth root; So, also, to find the value of a", we must first take the mth power of a, which is a", and then extract the nth root of that power: so that a" is the same as Va. Hence the numerator of a fractional exponent denotes the .power, and the denominator the root to be extracted. Again, let it be required to extract the cube root of 1 1 a In the first place, —=a. Now, to extract the cube root of a, we must divide its exponent by 3, which gives us 1 But the cube root of may also be represented by an 1 is equivalent to a a Thus we see that the principle of Art. 69, that a factor may be transferred from the numerator to the denominator of a traction, or from the denominator to the numerator by chang ing the sign of its exponent, is applicable also to fractional exponents. We may therefore entirely reject the radical signs hitherto made use of, and employ, in their stead, the fractional exponents which we have just explained; and, indeed, many of the difficulties in the reduction of radical quantities disappear when fractional exponents are substituted for the radical signs. PROBLEM I. To reduce surds to their most simple forms. (157.) Surds may frequently be simplified by the application of the following principle: the square root of the product of two or more factors is equal to the product of the square roots of those factors. Or, in algebraic language, √ab= vax√b. For each member of this equation squared will give the same quantity. Thus, the square of √ab is ab. And the square of √a× √b is (√a)3×(√b)2=ab. Hence, since the squares of the quantities ab and ✔ax√b are equal, the quantities themselves must be equal. Let it be required to reduce √4a to its most simple form. Hence, √4a=√4× √a=2√a=2aa. 2✓a is considered a simpler form than √4a, for reasons which will be better understood hereafter. Again, reduce 48 to its most simple form. ✓48 is equal to √16x3=√16× √3=4√3. Therefore, in order to simplify a monomial radical of the second degree, separate it into two factors, one of which is a perfect square; extract its root; and prefix it to the other factor with the radical sign between them. In the expressions 2a and 4/3, the quantities 2 and 4 are called the coefficients of the radical. EXAMPLES. 1. Reduce 2√32 to its most simple form. 2. Reduce √125a to its most simple form. 2 Reduce √98ab* to its most simple form. Ans. 8√2. Ans. 5a √5a Ans. 7b2 √2a. 4. Reduce √294ab2 to its most simple form. (158.) Surds of any degree may be simplified by the application of the following principle, which is merely an extension of that already proved in the preceding Article. The nth root of the product of any number of factors is equat to the product of the nth roots of those factors. Or, in algebraic language, Vab=Vax Vb. For, raise each of these expressions to the nth rower, and we shall obtain the same result. Thus, the nth power of Vab is ab. And the nth power of Vax Vb is (Va)"×(Vb)"= ab Hence, since the same powers of the quantities Vab ard Vax Vb are equal, the quantities themselves must be equal. Let it be required to reduce V8a to its most simple form. This is equivalent to V8X Va', which is equal to 2 Va2. Again, take the expression V48a. This is equivalent to 16a1× √3a, which is equal to 2a √3a. Hence, to simplify a monomial radical of any degree, we have the following RULE. Separate the quantity into two factors, one of which is an ex |