Root n-oh of degree from real number a is called such number b for which equality of b^n = is carried out by a. Roots of odd degree exist for negative and positive numbers, and roots of even degree - only for positive. Value of a root often is the recurring decimal decimal that complicates its exact calculation therefore it is important to be able to compare roots.
Instruction
1. Let it is required to compare two irrational numbers. The first what it is necessary to pay attention to is exponents of roots at the compared numbers. If indicators are identical, then compare radicands. It is obvious that the more the subradical number, the is more value of a root at equal indicators. For example, let it is necessary to compare a cubic root from two and a cubic root from eight. Indicators are identical and equal 3, radicands 2 and 8, and 2 <8. Therefore, and the cubic root from two is less than cubic root from eight.
2. In other case the exponents can be different, and radicands identical. Too it is quite clear that at extraction of a root of bigger degree the smaller number will turn out. Take for an example a cubic root from eight and a root of the sixth degree from eight. If to designate value of the first root as a, and the second - as b, then a^3 = 8 and b^6 = 8. It is easy to see that a has to be more b, thus the cubic root from eight is more than root of the sixth degree from eight.
3. The situation with different exponents of a root and different radicands is represented more difficult. In that case it is necessary to find a least common multiple for indicators of roots and build both expressions in the degree equal to a least common multiple. Example: it is necessary to compare 3^1/3 and 2^1/2 (mathematical record of roots is in the drawing). The least common multiple for 2 and 3 is equal to 6. Build both roots in the sixth degree. Right there it will turn out that 3^2 = 9 and 2^3 = 8, 9> 8. Therefore, and 3^1/3> 2^1/2.