Suppose that you want to compare two irrational numbers. The first thing you should pay attention to is the exponents of the roots of the numbers being compared. If the indicators are the same, compare radical expression. Obviously, the more radical the number, the greater the value of the root at equal rates. For example, suppose you need to compare the cube root of two and the cube root of eight. The indicators are the same and equal to 3, radical expressions 2 and 8 and 2 < 8. Consequently, the cube root of two less than the cube root of eight.
In another case, the exponents can be different, and radical expressions are the same. Also it is clear that the root more get fewer.Take for example the cube root of eight and the root of the sixth power of eight. If we denote the value of the first root as a and second as b, then a^3 = 8 and b^6 = 8. It is easy to see that a must be greater than b, so the cube root of eight more of the root of the sixth power of eight.
More complicated is the situation with different exponents roots and radical expressions different. In this case, it is necessary to find the least common multiple of the indices of the roots and build both expressions in a degree equal to the least common multiple.Example: need to compare 3^1/3 and 2^1/2 (mathematical notation has roots in the figure). Least common multiple for 2 and 3 is 6. Lift both the root of the sixth degree. There also turns out that 3^2 = 9 and 2^3 = 8, 9 > 8. Consequently, 3^1/3 > 2^1/2.
To compare arithmetic expressions composed of several roots, have lead to a common root. This can be done by using formulas of abridged multiplication formula of the Binomial theorem and other techniques.
Advice 2 : How to multiply square root square root
One of the four elementary mathematical operations (multiplication) has created another, somewhat more sophisticated exponentiation. That, in turn, has added additional complexity in the teaching of mathematics, giving rise to reverse a surgery - extraction of root. Any of these operations, you can use any other mathematical operation, which further confuses the study of the subject. To make it all somehow organize, there are sets of rules, one of which reglementary the order of multiplication of the roots.
Use for multiplication of square roots rule, the result should be the square root, radicals which is the product of radical expressions roots of multipliers. This rule applies when multiplying two, three or any other number of square roots. However, it applies not only to the roots of the quadratic, but cubic, or any other exponent, if this figure is the same for all involved in the operation of the radicals.
If the signs multiply roots are numerical values, multiply them together and place the resulting value under the root. For example, when multiplying √3,14 on √a 7.62 this action can be written as: √3,14 * √7,62 = √(3,14*7,62) = √23,9268.
If radical expressions contain variables, we first write the product under the same radical sign, and then try to simplify radical expression is obtained. For example, if it is necessary to multiply √(x+7) √(x-14) operation can be written as: √(x+7) * √(x-14) = √((x+7) * (x-14)) = √(x2-14*x+7*x-7*14) = √(x2-7*x-98).
If you need to multiply more than two square roots act in the same way - gather under one radical sign radical expression multiply all the roots as multipliers of one complex expression, and then simplify it. For example, when multiplying square roots of numbers 3,14, 7,62 and 5,56 can be written as: √3,14 * √7,62 * √5,56 = √(3,14*7,62*5,56) = √133,033008. And the multiplication of the square roots derived from expressions with variables x+7, x-14 and 2*x+1, so: √(x+7) * √(x-14) * √(2*x+1) = √((x+7) * (x-14) * (2*x+1)) = √((x2-14*x+7*x-7*14) * (2*x+1)) = √((x2-7*x-98) * (2*x+1)) = √(2*x*x2-2*x*7*x-2*x*98 + x2-7*x-98) = √(2*x3-14*x2-196*x+x2-7*x-98) = √(2*x3-13*x2-205*x-98).