Degree root n from number call such number which at construction in this degree will give that number from which the root is taken. Most often, actions are made with roots square which correspond 2 degrees. At extraction of a root it is often impossible to find it obviously, and the number which it is impossible to present in the form of natural fraction (transcendental) is result. But using some receptions, it is possible to simplify considerably the solution of examples with roots.
It is required to you
- - concept of a root from number;
- - actions with degrees;
- - formulas of abridged multiplication;
- - calculator.
Instruction
1. If absolute accuracy is not required, at the solution of examples with roots use the calculator. To take a square root from number, type it on the keyboard, and just press the corresponding button on which the sign of a root is represented. As a rule, on calculators the root square undertakes. But for calculation of roots of the highest degrees, use function of construction of number in degree (on the engineering calculator).
2. For extraction of a square root build number in degree 1/2, a cubic root in 1/3 and so on. At the same time surely consider that at extraction of roots of even degrees, the number has to be positive, otherwise the calculator just will not issue the answer. It is connected with the fact that at construction in even degree any number will be positive, for example, (-2) ^4= (-2) ∙ (-2) ∙ (-2) ∙ (-2)=16. For extraction of a square root totally when it is possible, use the table of squares of natural numbers.
3. If there is no calculator nearby, or absolute accuracy in calculations is required, use properties of roots and also various formulas for simplification of expressions. It is possible to take a root from many numbers partially. For this purpose use property that the root from the work of two numbers is equal to the work of roots from these numbers √m∙n= √ to m ∙√ to n.
4. Example. Calculate value of expression (√80-√ 45) / √5. Direct calculation will give nothing as any root totally is not taken. Transform expression (√16∙5-√ 9∙5) / √5= (√ 16 ∙√ 5-√ 9 ∙√ 5) / √5= √ 5 ∙ (√ 16-√ 9) / √5. Make reduction of numerator and a denominator on √5, receive (√16-√ 9)=4-3=1.
5. If the radicand or a root are built in degree, then at extraction of a root use that property that the exponent of a radicand can be divided into root degree. If division is made totally, the number is brought from under a root. For example, √5^4=5²=25. Example. To calculate value of expression (√3+ √ 5) ∙ (√3-√ 5). Apply a formula of a difference of squares and receive (√3)²-(√ 5)²=3-5=-2.