2. Find the cube root of 39651821. Ans. 341. Ans. 159. 211. Cube Root of Fractions.-The cube root of a fraction is equal to the root of its numerator divided by the root of its denominator. Hence the cube root of 343 is The number 12.167 may be written 127, and its cube root is 23, or 2.3. That is, the cube root of a decimal fraction, or of a whole number followed by a decimal fraction, may be found in the same manner as that of a whole number, if we divide it into periods commencing with the decimal point. In the extraction of the cube root of an integer, if there is still a remainder after we have obtained the units' figure of the root, it indicates that the proposed number has not an exact cube root. We may, if we please, proceed with the approximation to any desired extent, by supposing a decimal point at the end of the proposed number, and annexing any number of periods of three ciphers each, and continuing the operation. We thus obtain a decimal part to be added to the integral part already found. So, also, if a decimal number has no exact cube root, we may annex ciphers, and proceed with the approximation to any desired extent. 2. Find the cube root of 14!!. 3. Find the cube root of 13.312053. 4. Find the cube root of 1892.819053. 5. Find the cube root of .001879080904. Ans. 18. Ans. 23. Find the cube roots of the following numbers to 5 decimal CHAPTER XIII. RADICAL QUANTITIES. 212. A radical quantity is an indicated root of a quantity: as va, Va, etc. Radical quantities may be either surd or rational. Radical quantities are divided into degrees, the degree being denoted by the index of the root. Thus, √3 is a radical of the second degree; V5 is a radical of the third degree, etc. 213. The coefficient of a radical is the number or letter prefixed to it, showing how often the radical is to be taken. Thus, in the expression 2 √a, 2 is the coefficient of the radical. Similar radicals are those which have the same index and the same quantity under the radical sign. Thus, 3√ã and 5√ā are similar radicals. Also 7 and 10 are similar radicals. 214. Use of fractional Exponents.-We have seen, Art. 196, that in order to extract any root of a monomial, we must divide the exponent of each literal factor by the index of the required root. Thus the square root of a1 is a2, and in the same manner the square root of a3 may be written a*, that of a5 will be a3, and that of a, or a', is a. Whence we see that So, also, the cube root of a2 may be written a3; the cube root of a1 is a3; and the cube root of a, or a1, is a1. Whence we see that aR is equivalent to Va, a 3 In the same manner, at is equivalent to va, That is, the numerator of a fractional exponent denotes the power, and the denominator the root to be extracted. Let it be required to extract the cube root of 1 This quan tity, Art. 187, is equivalent to a-4. Now, to extract the cube root of a 4, we must divide its exponent by 3, which gives us But the cube root of may also be represented by 1 Thus we see that the principle of Art. 77, that a factor may be transferred from the numerator to the denominator of a fraction, or from the denominator to the numerator by changing the sign of its exponent, is applicable also to fractional exponents. We may therefore entirely reject the radical signs hitherto employed, and substitute for them fractional exponents, and many of the difficulties which occur in the reduction of radical quantities are thus made to disappear. To reduce a Radical to its simplest Form. 215. A radical is in its simplest form when it has under the radical sign no factor which is a perfect power corresponding to the degree of the radical. Radical quantities may frequently be simplified by the application of the following principle: the nth root of the product of two or more factors is equal to the product of the nth roots of those factors; or, in algebraic language, Vab=VaxVb. For each of these expressions, raised to the nth power, will give the same quantity. Thus, the nth power of Vab is ab. And the nth power of Va× Vb is (Vā)" × (Võ)", or ab. Hence, since the same powers of the quantities Vab and VaxVb are equal, the quantities themselves must be equal. Let it be required to reduce √48a3x2 to its simplest form. This expression may be put under the form √16a2x2× √3a But √16a2x2 is equal to 4ɑx. Hence, to reduce a radical to its simplest form, we have the following RULE. Resolve the quantity under the radical sign into two factors, one of which is the greatest perfect power corresponding in degree to the radical. Extract the required root of this factor, and prefix it to the other factor, which must be left under the sign. 216. When the quantity under the radical sign is a fraction, it is often convenient to multiply both its terms by such a quantity as will make the denominator a perfect power of the degree indicated. Then, after simplifying, the factor remaining under the radical sign will be entire. 217. The following principle can frequently be employed in simplifying radicals: The mnth root of any quantity is equal to the mth root of the nth root of that quantity. That is, Ans. Vabc. Ans. VA+VI. Ans. V9+1V18. |