Actually, the square root (√) is only the symbol designating exponentiation ½. Therefore when finding a square root from the number or expression built in some degree it is possible to use usual rules of "construction of degree in degree". It is necessary to consider some nuances only.
It is required to you
- - calculator;
- - paper;
- - pencil.
Instruction
1. To find a square root from degree of non-negative number, just increase a radicand exponent on ½ (or divide into 2). Example. √ (2²) = 2^ (½ * 2) = 2^1 = 2 (^ - a badge of exponentiation). √ (x²) = x ^ (½ * 2) = x^1 = x, for all h0.
2. If the radicand can accept negative values, then the aforesaid governed use carefully. As the square root from a negative number is not defined (if not to press in the area of complex numbers), then exclude such intervals from a function range of definition. Though kh and х^½ - ½ it is very easy "to lose" equivalent expressions, an exponent at further transformations.
3. If the squared expression can accept negative values, then use the following formula: kh² = |x|, where |x| - standard designations of the module (absolute value) of number. So, for example, √ (-1)² = |-1| = 1 Apply the Similar rule when degree is even number. √ (х^(2n)) = |x^n| where n is an integer.
4. Finding of a range of definition of the "square root" function is often much more difficult than calculation of the value of function. If under the sign of a square root some expression of X is located, then solve inequality X0.
5. Consider that as kh² = |x|, from squares of two numbers does not follow from equality of roots at all that numbers are equal. This nuance is often used for the invention of all funny "proofs" of type 2=3 or 2*2=5. Therefore attentively you carry out all transformations with similar expressions. By the way, such tasks quite often meet in examination tasks, and the task can have very indirect relation to extraction of roots (for example, trigonometrical expressions or derivatives).