Instruction

1

First, the addition of square roots, try to remove these roots. This will be possible if the number under the root sign are complete squares. For example, suppose you specify an expression √4 + √9. The first number 4 is the square of the number 2. Second the number 9 is the square of the number 3. So it turns out that: √4 + √9 = 2 + 3 = 5.

2

If under the root sign has no full squares, then try to stand under the sign of the root of the multiplier number. For example, suppose that the expression √24 + √54. Decompose numbers into factors: 24 = 2 * 2 * 2 * 3, 54 = 2 * 3 * 3 * 3. Among the 24, a multiplier 4, which can be taken from under the square root sign. Among the 54 - multiplier 9. Thus, it appears that: √24 + √54 = √(4 * 6) + √(9 * 6) = 2 * √6 + 3 * √6 = 5 * √6. In this example, as a result of the removal of the multiplier from under the sign of the root has the goal to simplify the given expression.

3

Let the sum of the two square roots is the denominator of a fraction, e.g., A / (√a + √b). And let your goal is "get rid of irrationality in denominator". Then you can use the following method. Multiply the numerator and denominator in the expression √a - √b. Thus the denominator will be a reduced multiplication formula: (√a + √b) * (√a - √b) = a – b. Similarly, if the denominator is given the difference of the roots: √a - √b, then the numerator and denominator must be multiplied by the expression √a + √b. For example, given a fraction 4 / (√3 + √5) = 4 * (√3 - √5) / ( (√3 + √5) * (√3 - √5) ) = 4 * (√3 - √5) / (-2) = 2 * (√5 - √3).

4

Consider a more complex example getting rid of irrationality in denominator. Let the given fraction 12 / (√2 + √3 + √5). You need to multiply the numerator and denominator in the expression √2 + √3 - √5:

12 / (√2 + √3 + √5) = 12 * (√2 + √3 - √5) / ( (√2 + √3 + √5) * (√2 + √3 - √5) ) = 12 * (√2 + √3 - √5) / (2 * √6) = √6 * (√2 + √3 - √5) = 2 * √3 + 3 * √2 - √30.

12 / (√2 + √3 + √5) = 12 * (√2 + √3 - √5) / ( (√2 + √3 + √5) * (√2 + √3 - √5) ) = 12 * (√2 + √3 - √5) / (2 * √6) = √6 * (√2 + √3 - √5) = 2 * √3 + 3 * √2 - √30.

5

And finally, if you only need an approximate value, it is possible to calculate the values of square roots on the calculator. Calculate values separately for each number and record with the required accuracy (for example, two decimal places). And then take the required arithmetic operations as ordinary numbers. For example, suppose you want to know the approximate value of the expression √7 + √5 ≈ 2,65 + 2,24 = 4,89.

Note

Square roots in any case can not be folded as a simple number, ie.√3 + √2 ≠ √5!!!

Useful advice

If you decompose a number into factors to make a square under the root sign, then take the opposite test - multiply all the multipliers and get the original number.

# Advice 2 : How to calculate the square root

Calculating the square roots scared some students at first. Let's see how they need to work and what to pay attention to. Also we give their properties.

Instruction

1

About the use of the calculator will not speak, although, of course, in many cases it is simply necessary.

So, the square root of the number x is the number y, which squared gives the number x.

Definitely need to remember one very important point: the square root is computed only from positive numbers (not complex take). Why? See the definition written above. The second important point: the result of the root extraction if there are no additional conditions in the General case there are two integers: +y and -y (in the General case, the module y), as both of them squared gives the original number x, which does not contradict the definition.

The square root of zero is zero.

So, the square root of the number x is the number y, which squared gives the number x.

Definitely need to remember one very important point: the square root is computed only from positive numbers (not complex take). Why? See the definition written above. The second important point: the result of the root extraction if there are no additional conditions in the General case there are two integers: +y and -y (in the General case, the module y), as both of them squared gives the original number x, which does not contradict the definition.

The square root of zero is zero.

2

Now, with regard to specific examples. For small numbers the squares (and hence the roots as the inverse operation) it is best to remember the multiplication table. I'm talking about the numbers from 1 to 20. This will save your time and help in assessing the possible value of the desired root. For example, knowing that the root of 144 = 12, and the root of 13 = 169, we can estimate that the root of the number 155 is between 12 and 13. A similar evaluation can be applied for larger numbers, their only difference is the complexity and execution time of these operations.

Also, there is another simple fun way. Let's show it on example.

Suppose there is a number 16. Find out which number is his root. To do this, we successively subtract of the 16 primes and count the number of operations performed.

So, 16-1=15 (1), 15-3=12 (2), 12-5=7 (3), 7-7=0 (4). 4 operations – the required number 4. The essence is to carry out the subtraction as long as the difference does not become equal to 0 or will simply be deducted less than the next Prime number.

The disadvantage of this method is that thus it is possible to find only the integer part of the root, but not all of its exact meaning completely, but sometimes with a precision of estimation or errors of computation, and this is enough.

Also, there is another simple fun way. Let's show it on example.

Suppose there is a number 16. Find out which number is his root. To do this, we successively subtract of the 16 primes and count the number of operations performed.

So, 16-1=15 (1), 15-3=12 (2), 12-5=7 (3), 7-7=0 (4). 4 operations – the required number 4. The essence is to carry out the subtraction as long as the difference does not become equal to 0 or will simply be deducted less than the next Prime number.

The disadvantage of this method is that thus it is possible to find only the integer part of the root, but not all of its exact meaning completely, but sometimes with a precision of estimation or errors of computation, and this is enough.

3

Some basic properties: the root of the sum (difference) is not equal to the sum (difference) of the roots, but the root of the product of (private) equal to the product of (private) roots.

The root of the square of the number x is the number x.

The root of the square of the number x is the number x.

# Advice 3 : How to calculate square root of a number

The operation calculates the root of any number means to find the value at which the multiplication of this value on yourself as many times as indicated in the figure of the root, resulting in a radical number. If the index of the root is equal to two, such a root is called "square". If you need to calculate the square root of the computer user have a choice from several options.

Instruction

1

Use any calculator to calculate the roots with the exponent equal to two ("square"). A link to the software running the calculator can be found in the section "business" subsection "Standard" under "All programs" in the main menu of Windows. To calculate the square root of the interface button marked with symbols, sqrt (SQuare RooT - square root).

2

Use the calculator built into the search engine Nigma or Google, if you need the easiest way to figure out the value of the square root of any number. For example, if you want to calculate the root of a number 989, then go to the home page of any of these search engines and enter "root of 989". The designation used in the calculator Windows to denote this operation, you can apply here and request sqrt 989 will also be processed correctly by the search engine.

3

Use the built-in function ROOT, if you can attract to solve the problem spreadsheet editor of Microsoft Word Excel. To do this, run the application and enter in the first cell of the radical number. Then go to the cell in which you want to see the result of the calculation, and click the "Insert function" - it is placed above the table, to the left of the formula bar.

4

In the dialog that opens, select "Math" in the drop-down Category list, and then click in the list of the functions of the ROOT, and then click OK. In the window "function Arguments" select the cell with radical numbers - just click on it with the mouse. Click OK - Excel will calculate and display the value of the square root. You will then have the opportunity to change radical number, and the cell with the formula will display a value of a square root to a new value.

# Advice 4 : How to build a root square

The construction of the degree is the abbreviated form of record of the operation of repeated multiplication, in which all multipliers is equal to the original number. And the root means - the definition of the multiplier that should be involved in the operation of multiple multiplication that its result is a radical number. As the exponent and the index of the root point to the same thing - how many factors should be a multiplication operation.

You will need

- Access to the Internet.

Instruction

1

If the number or expression you want to apply simultaneously, and the operation of root extraction, and the construction of its degree, bring both actions in a single - exponentiation with a fractional exponent. In the numerator of the fraction must be the exponent, and the denominator the root. For example, if you want to build in

**the square**of cubic**root**, then the two operations will be equivalent to one construction of the power⅔.2

If the conditions required to build in

**square****root**with exponent equal to two, this task is not the calculation, and to check your knowledge. Use method from the first step and you will get the fraction 2/2, i.e. 1. This means that the result of raising to**the square****the square**tion of the square root of any number is the number itself.3

If necessary, raised to

**a square****root**with an even exponent, it is always possible to simplify the operation. Since the two (the numerator of the fractional exponent) and any even number (the denominator) have a common divisor, then after simplification of the fraction in the numerator will be one and that means to involve in calculations, it is enough to extract**the root**with half the exponent. For example, in the construction of**the square**root, the sixth power of eight can be reduced to extracting from it a cube root, because 2/6=1/3.4

To calculate the result in all indicators of the degree of root use, for example, a calculator built into the Google search engine. This is perhaps the easiest method of payment you have access to the Internet from your computer. A common substitute symbol for the operation of exponentiation is that such a "lid": ^. Use it when typing in Google search query. For example, if you want to build in

**square****root**of the fifth degree from among 750, formulate the query: 750^(2/5). After entering the search without even clicking send to the server will show the result of calculations with the accuracy of seven decimal places: 750^(2 / 5) = 14,1261725.# Advice 5 : How do I output the number of root

**The number**that 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.

Instruction

1

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.

2

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.

3

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.

4

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.

5

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).

6

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.