Use the definition of rootas a mathematical operation, from which it follows that the extraction of a root is the inverse operation of raising a number to a power. This means that the number can be taken from under the root if the decrease in radical expressions , the number of times that corresponds to a raised to the power passed. For example, to take from under the square root of the number 10, divide remaining under the root expression of ten squared.
Pick up the radical number of this multiplier, making of which is under root do simplify the expression - otherwise the operation will lose meaning. For example, if under the sign of the root with exponent equal to three (cubic root), is the number 128, then out of sign can be taken, for example, the number 5. In this radical , the number 128 will have to be divided by 5 cubed: 3√128 = 5∗3√(128/53) = 5∗3√(128/125) = 5∗3√1.024. If the presence of fractional numbers under the sign of root is not contrary to the conditions of the problem, the solution can be left in this form. If you need a more simple variant, we first divide radical expression for such an integer multiplier, cube root one of which will be a whole numberC. for Example: 3√128 = 3√(64∗2) = 3√(43∗2) = 4∗3√2.
Use for the selection of the multipliers radical number calculator, if you calculate in the mind the powers of a number is not possible. This is especially important for the rootm with the exponent greater than two. If you have Internet access, you can calculate the built-in Google search engine and Nigma solvers. For example, if you need to find the largest integer multiplier, which can be taken from under the sign of the cubic root for numbers 250, then going to Google enter "6^3" to check, can you just take out of sign of root six. The search engine displays the result equal to 216. Alas, the 250 cannot be divided without a remainder is the number. Then enter the query 5^3. The result will be 125, and it allows you to break 250 on the multipliers 125 and 2, and therefore to stand under the sign of the root number 5, leaving the number 2.
Advice 2: How to make a multiplier from under the sign of the root
To take out from under the root of one of the factors necessary in situations when it is necessary to simplify a mathematical expression. There are times when to perform the desired calculations using the calculator it is impossible. For example, if instead of numbers uses letters denote variables.
Lay out radical expression in simple factors. Let's see which of the factors is repeated as many times as indicated in the figures of the root, or more. For example, you need to extract the cubic root of a number and fourth degree. In this case, the number can be represented as a*a*a*a = a*(a*a*a)=a*A3. The index of the root in this case will correspond withthe multiplier A3. And it should be taken out of the radical sign.
Summarize the properties of the roots. The imposition of under the sign of the radical is an action that is the opposite of exponentiation. That is, in this case, it is necessary to extract the cube root of that part of the expression that yields this operation, which in this case is A3 3√a*a3 =a3√a.
Check out the calculations. This is especially important if you are dealing with numbers, not with lettered variables. For example, you need to convert the expression 3√120. Expanding radical expression into a fraction, you get 3√120=3√(60*2)=3√(30*2*2)=3√(15*2*2*2)=3√(3*5*2*2*2). Under root you can make witha multiplier of 2. Will get 23√15. Check the result. To do this, make the multiplier of the root, previously elevating him to the appropriate degree. 23 = 8. Accordingly, 23√15 = 3√(15*8) = 3√120.
For the decomposition into simple factors of numbers with lots of digits use a calculator. It is useful to do with roote, the rate of which is greater than two. When working with variables marked is not so important, since accurate calculations are not needed.
Use the search engines. This is necessary, for example, to find the highest integer multiplier, which can be taken from under the sign of the radical. Use Nigma. In the search engine type the number and what to do with it. For example, enter the expression "120 to factorize". You will receive a response 23 (3*5), that is the same thing that you have achieved through oral calculations in the given example. If you need a precise calculation, use the online calculator.
The imposition of a multiplier from under the root makes sense only if this action really simplifies the expression.
Advice 3: How do I output the number of root
The numberthat is under the sign of the root, often hinders the solution of the equation, it is inconvenient to work with. Even if it is raised to a power, fractional or not may be provided in whole numbers to a certain extent, you can try to deduce it from the root, completely or at least partially.
Try to decompose the number into Prime factors. If the number is decimal, not a comma yet, consider, consider all the numbers. For example, the number of 8.91 can be expanded as: of 8.91=0,9*0,9*11 (first place 891=9*9*11 then add the commas). Now you can record the number of 0.9^2*11 and to withdraw from the root of 0.9. So you got √of 8.91=0,9√11.
If you are given a cube root, you need to display underneath the number to the third power. For example, the number 135 decompose as 3*3*3*5=3^3*5. From under the root, print the number 3, the number 5 will remain under the root sign. Exactly the same goes with the roots of the fourth and higher degrees.
To bring out the root of the number with a degree other than the degree of the root (e.g., square root, and under it the number in grade 3), do so. Note the root degree, that is, put the sign √ and put in place a sign degree. For example, the square root of a number equals the same number of degree½, and a cubic – to the power of 1/3. Do not forget to enclose the radical expression in parenthesis.
Simplify the expression multiplying degree. For example, if the root was 12^4, and the root was square, the expression takes the form (12^4)^1/2=12^4/2=12^2=144.
To withdraw from the sign of the root is possible and a negative number. If the degree is odd, just imagine the number under the root as a number in the same way, for example, -8=(-2)^3, the cube root of (-8) is equal to (-2).
To make a negative number under an even root of a degree (including square), do so. Imagine radical expression as a product of (-1) and the number in the right degree, then take the number leaving (-1) under the root sign. For example, √(-144)=√(-1)*√144=12*√(-1). The number √(-1) in mathematics called imaginary numbers, and denoted by parameter i. Thus, √(-144)=12i.