Engineering calculator online

We are happy to present everyone with a free engineering calculator. With its help, any student can quickly and, most importantly, easily perform various types of mathematical calculations online.

The calculator is taken from the site - web 2.0 scientific calculator

A simple and easy-to-use engineering calculator with an unobtrusive and intuitive interface will truly be useful to a wide range of Internet users. Now, whenever you need a calculator, go to our website and use the free engineering calculator.

An engineering calculator can perform both simple arithmetic operations and quite complex mathematical calculations.

Web20calc is an engineering calculator that has a huge number of functions, for example, how to calculate all elementary functions. The calculator also supports trigonometric functions, matrices, logarithms and even plotting.

Undoubtedly, Web20calc will be of interest to that group of people who are looking for simple solutions dials in search engines request: online mathematical calculator. A free web application will help you instantly calculate the result of any mathematical expression, for example, subtract, add, divide, take the root, raise to a power, etc.

In the expression, you can use the operations of exponentiation, addition, subtraction, multiplication, division, percentage, and the PI constant. For complex calculations, parentheses should be included.

Features of the engineering calculator:

1. basic arithmetic operations;
2. working with numbers in a standard form;
3. calculation trigonometric roots, functions, logarithms, exponentiation;
4. statistical calculations: addition, arithmetic mean or standard deviation;
5. use of memory cells and custom functions of 2 variables;
6. work with angles in radian and degree measures.

The engineering calculator allows the use of a variety of mathematical functions:

Extracting roots (square, cubic, and nth root);
ex (e to the x power), exponential;
trigonometric functions: sine - sin, cosine - cos, tangent - tan;
inverse trigonometric functions: arcsine - sin-1, arccosine - cos-1, arctangent - tan-1;
hyperbolic functions: sine - sinh, cosine - cosh, tangent - tanh;
logarithms: binary logarithm to base two - log2x, decimal logarithm base ten - log, natural logarithm - ln.

This engineering calculator also includes a value calculator with the ability to convert physical quantities For various systems measurements - computer units, distance, weight, time, etc. Using this function, you can instantly convert miles to kilometers, pounds to kilograms, seconds to hours, etc.

To make mathematical calculations, first enter a sequence of mathematical expressions in the appropriate field, then click on the equal sign and see the result. You can enter values ​​directly from the keyboard (for this, the calculator area must be active, therefore, it would be useful to place the cursor in the input field). Among other things, data can be entered using the buttons of the calculator itself.

To build graphs, you should write the function in the input field as indicated in the field with examples or use the toolbar specially designed for this (to go to it, click on the button with the graph icon). To convert values, click Unit; to work with matrices, click Matrix.

We have already sorted out a large number without a calculator. In this article we will look at how to extract the cube root (third degree root). Let me make a reservation that we are talking about natural numbers. How long do you think it takes to verbally calculate roots such as:

Quite a bit, and if you practice two or three times for 20 minutes, then you can extract any such root orally in 5 seconds.

*It should be noted that we are talking about numbers under the root that are the result of cubed natural numbers from 0 to 100.

We know that:

So, the number a that we will find is natural number from 0 to 100. Look at the table of cubes of these numbers (results of raising to the third power):

You can easily extract the cube root of any number in this table. What do you need to know?

1. These are cubes of numbers that are multiples of ten:

I would even say that these are “beautiful” numbers, they are easy to remember. It's easy to learn.

2. This is a property of numbers when they are producted.

Its essence lies in the fact that when a certain number is raised to the third power, the result will have a peculiarity. Which one?

For example, let's cube 1, 11, 21, 31, 41, etc. You can look at the table.

1 3 = 1, 11 3 = 1331, 21 3 = 9261, 31 3 = 26791, 41 3 = 68921 …

That is, when we cube a number with a unit at the end, the result will always be a number with a unit at the end.

When you cube a number with a two at the end, the result will always be a number with an eight at the end.

Let's show the correspondence in the table for all numbers:

Knowledge of the two points presented is quite enough.

Let's look at examples:

Take the cube root of 21952.

This number is in the range from 8000 to 27000. This means that the result of the root lies in the range from 20 to 30. The number 29952 ends in 2. This option is only possible when a number with an eight at the end is cubed. Thus, the result of the root is 28.

Find the cube root of 54852.

This number is in the range from 27000 to 64000. This means that the result of the root lies in the range from 30 to 40. The number 54852 ends in 2. This option is only possible when a number with an eight at the end is cubed. Thus, the result of the root is 38.

Take the cube root of 571787.

This number is in the range from 512000 to 729000. This means that the result of the root lies in the range from 80 to 90. The number 571787 ends in 7. This option is only possible when a number with a three at the end is cubed. Thus, the result of the root is 83.

Take the cube root of 614125.

This number is in the range from 512000 to 729000. This means that the result of the root lies in the range from 80 to 90. The number 614125 ends in 5. This option is only possible when a number with a five at the end is cubed. Thus, the result of the root is 85.

I think that you can now easily extract the cube root of the number 681472.

Of course, extracting such roots orally takes a little practice. But by restoring the two indicated tablets on paper, you can easily extract such a root within a minute, in any case.

After you have found the result, be sure to check it (raise it to the third power). *Nobody canceled multiplication by column 😉

Actually Unified State Exam problems with such “ugly” roots, no. For example, you need to extract the cube root of 1728. I think this is no longer a problem for you.

If you know any interesting methods of calculations without a calculator, send them, I will publish them in due course.That's all. Good luck to you!

Sincerely, Alexander Krutitskikh.

P.S: I would be grateful if you tell me about the site on social networks.

Posted on our website. Taking the root of a number is often used in various calculations, and our calculator is an excellent tool for such mathematical calculations.

An online calculator with roots will allow you to quickly and easily make any calculations involving root extraction. The third root can be calculated as easily as the square root of a number, the root of negative number, root of a complex number, root of pi, etc.

Calculating the root of a number is possible manually. If it is possible to calculate the whole root of a number, then we simply find the value of the radical expression using the table of roots. In other cases, the approximate calculation of roots comes down to decomposing the radical expression into a product of simpler factors, which are powers and can be removed by the sign of the root, simplifying the expression under the root as much as possible.

But you shouldn't use this root solution. And here's why. Firstly, you will have to spend a lot of time on such calculations. Numbers at the root, or more precisely, expressions can be quite complex, and the degree is not necessarily quadratic or cubic. Secondly, the accuracy of such calculations is not always satisfactory. And thirdly, there is an online root calculator that will do any root extraction for you in a matter of seconds.

To extract a root from a number means to find a number that, when raised to the power n, will be equal to the value of the radical expression, where n is the power of the root, and the number itself is the base of the root. The root of the 2nd degree is called simple or square, and the root of the third degree is called cubic, omitting the indication of the degree in both cases.

Solving the roots in online calculator comes down to just writing a mathematical expression in the input line. Extracting from a root in the calculator is designated as sqrt and is performed using three keys - extract square root sqrt(x), cube root sqrt3(x) and nth root sqrt(x,y). More detailed information about the control panel is presented on the page.

Square Root

Clicking this button will insert the square root entry in the input line: sqrt(x), you only need to enter the radical expression and close the parenthesis.

An example of solving square roots in a calculator:

If the root is a negative number and the degree of the root is even, then the answer will be represented as a complex number with imaginary unit i.

Square root of a negative number:

Third root

Use this key when you need to take the cube root. It inserts the entry sqrt3(x) into the input line.

3rd degree root:

Root of degree n

Naturally, the online roots calculator allows you to extract not only the square and cubic roots of a number, but also the root of degree n. Clicking this button will display an entry like sqrt(x x,y).

4th root:

An exact nth root of a number can only be extracted if the number itself is an exact nth root. Otherwise, the calculation will turn out to be approximate, although very close to ideal, since the accuracy of the online calculator’s calculations reaches 14 decimal places.

5th root with approximate result:

Root of a fraction

The calculator can calculate the root from various numbers and expressions. Finding the root of a fraction comes down to separately extracting the root of the numerator and denominator.

Square root of a fraction:

Root from the root

In cases where the root of the expression is under the root, according to the properties of roots, they can be replaced by one root, the degree of which will be equal to the product of the degrees of both. Simply put, to extract a root from a root, it is enough to multiply the indicators of the roots. In the example shown in the figure, the expression third-degree root of the second-degree root can be replaced by one 6th-degree root. Specify the expression as you wish. In any case, the calculator will calculate everything correctly.

An example of how to extract a root from a root:

Degree at the root

The root of the degree calculator allows you to calculate in one step, without first reducing the root and degree indicators.

Square root of a degree:

All functions of our free calculator are collected in one section.

Solving roots in an online calculator was last modified: March 3rd, 2016 by Admin

It's time to sort it out root extraction methods. They are based on the properties of roots, in particular, on the equality, which is true for any non-negative number b.

Below we will look at the main methods of extracting roots one by one.

Let's start with the simplest case - extracting roots from natural numbers using a table of squares, a table of cubes, etc.

If tables of squares, cubes, etc. If you don’t have it at hand, it’s logical to use the method of extracting the root, which involves decomposing the radical number into prime factors.

It is worth special mentioning what is possible for roots with odd exponents.

Finally, let's consider a method that allows us to sequentially find the digits of the root value.

Let's get started.

Using a table of squares, a table of cubes, etc.

In the simplest cases, tables of squares, cubes, etc. allow you to extract roots. What are these tables?

The table of squares of integers from 0 to 99 inclusive (shown below) consists of two zones. The first zone of the table is located on a gray background; by selecting a specific row and a specific column, it allows you to compose a number from 0 to 99. For example, let's select a row of 8 tens and a column of 3 units, with this we fixed the number 83. The second zone occupies the rest of the table. Each cell is located at the intersection of a certain row and a certain column, and contains the square of the corresponding number from 0 to 99. At the intersection of our chosen row of 8 tens and column 3 of ones there is a cell with the number 6,889, which is the square of the number 83.

Tables of cubes, tables of fourth powers of numbers from 0 to 99, and so on are similar to the table of squares, only they contain cubes, fourth powers, etc. in the second zone. corresponding numbers.

Tables of squares, cubes, fourth powers, etc. allow you to extract square roots, cube roots, fourth roots, etc. accordingly from the numbers in these tables. Let us explain the principle of their use when extracting roots.

Let's say we need to extract the nth root of the number a, while the number a is contained in the table of nth powers. Using this table we find the number b such that a=b n. Then , therefore, the number b will be the desired root of the nth degree.

As an example, let's show how to use a cube table to extract the cube root of 19,683. We find the number 19,683 in the table of cubes, from it we find that this number is the cube of the number 27, therefore, .

It is clear that tables of nth powers are very convenient for extracting roots. However, they are often not at hand, and compiling them requires some time. Moreover, it is often necessary to extract roots from numbers that are not contained in the corresponding tables. In these cases, you have to resort to other methods of root extraction.

Factoring a radical number into prime factors

Enough in a convenient way, which makes it possible to extract a root from a natural number (if, of course, the root is extracted), is the decomposition of the radical number into prime factors. His the point is this: after that it is quite easy to represent it as a power with the desired exponent, which allows you to obtain the value of the root. Let's clarify this point.

Let the nth root of a natural number a be taken and its value equal b. In this case, the equality a=b n is true. The number b, like any natural number, can be represented as the product of all its prime factors p 1 , p 2 , …, p m in the form p 1 ·p 2 ·…·p m , and the radical number a in this case is represented as (p 1 ·p 2 ·…·p m) n . Since the decomposition of a number into prime factors is unique, the decomposition of the radical number a into prime factors will have the form (p 1 ·p 2 ·…·p m) n, which makes it possible to calculate the value of the root as.

Note that if the decomposition into prime factors of a radical number a cannot be represented in the form (p 1 ·p 2 ·…·p m) n, then the nth root of such a number a is not completely extracted.

Let's figure this out when solving examples.

Example.

Take the square root of 144.

Solution.

If you look at the table of squares given in the previous paragraph, you can clearly see that 144 = 12 2, from which it is clear that the square root of 144 is equal to 12.

But in light of this point, we are interested in how the root is extracted by decomposing the radical number 144 into prime factors. Let's look at this solution.

Let's decompose 144 to prime factors:

That is, 144=2·2·2·2·3·3. Based on the resulting decomposition, the following transformations can be carried out: 144=2·2·2·2·3·3=(2·2) 2·3 2 =(2·2·3) 2 =12 2. Hence, .

Using the properties of the degree and the properties of the roots, the solution could be formulated a little differently: .

Answer:

To consolidate the material, consider the solutions to two more examples.

Example.

Calculate the value of the root.

Solution.

The prime factorization of the radical number 243 has the form 243=3 5 . Thus, .

Answer:

Example.

Is the root value an integer?

Solution.

To answer this question, let's factor the radical number into prime factors and see if it can be represented as a cube of an integer.

We have 285 768=2 3 ·3 6 ·7 2. The resulting expansion cannot be represented as a cube of an integer, since the power of the prime factor 7 is not a multiple of three. Therefore, the cube root of 285,768 cannot be extracted completely.

Answer:

No.

Extracting roots from fractional numbers

It's time to figure out how to extract the root from fractional number. Let the fractional radical number be written as p/q. According to the property of the root of a quotient, the following equality is true. From this equality it follows rule for extracting the root of a fraction: The root of a fraction is equal to the quotient of the root of the numerator divided by the root of the denominator.

Let's look at an example of extracting a root from a fraction.

Example.

What is the square root of common fraction 25/169 .

Solution.

Using the table of squares, we find that the square root of the numerator of the original fraction is equal to 5, and the square root of the denominator is equal to 13. Then . This completes the extraction of the root of the common fraction 25/169.

Answer:

The root of a decimal fraction or mixed number is extracted after replacing the radical numbers with ordinary fractions.

Example.

Take the cube root of the decimal fraction 474.552.

Solution.

Let's imagine the original decimal as a common fraction: 474.552=474552/1000. Then . It remains to extract the cube roots that are in the numerator and denominator of the resulting fraction. Because 474 552=2·2·2·3·3·3·13·13·13=(2 3 13) 3 =78 3 and 1 000 = 10 3, then And . All that remains is to complete the calculations .

Answer:

.

Taking the root of a negative number

It is worthwhile to dwell on extracting roots from negative numbers. When studying roots, we said that when the root exponent is an odd number, then there can be a negative number under the root sign. We gave these entries the following meaning: for a negative number −a and an odd exponent of the root 2 n−1, . This equality gives rule for extracting odd roots from negative numbers: to extract the root of a negative number, you need to take the root of the opposite positive number, and put a minus sign in front of the result.

Let's look at the example solution.

Example.

Find the value of the root.

Solution.

Let's transform the original expression so that there is a positive number under the root sign: . Now mixed number replace it with an ordinary fraction: . We apply the rule for extracting the root of an ordinary fraction: . It remains to calculate the roots in the numerator and denominator of the resulting fraction: .

Here is a short summary of the solution: .

Answer:

.

Bitwise determination of the root value

IN general case under the root there is a number that, using the techniques discussed above, cannot be represented as the nth power of any number. But at the same time there is a need to know the meaning given root, at least up to a certain sign. In this case, to extract the root, you can use an algorithm that allows you to sequentially obtain a sufficient number of digit values ​​of the desired number.

The first step of this algorithm is to find out what the most significant bit of the root value is. To do this, the numbers 0, 10, 100, ... are sequentially raised to the power n until the moment when a number exceeds the radical number is obtained. Then the number that we raised to the power n at the previous stage will indicate the corresponding most significant digit.

For example, consider this step of the algorithm when extracting the square root of five. Take the numbers 0, 10, 100, ... and square them until we get a number greater than 5. We have 0 2 =0<5 , 10 2 =100>5, which means the most significant digit will be the ones digit. The value of this bit, as well as the lower ones, will be found in the next steps of the root extraction algorithm.

All the following steps of the algorithm are aimed at sequentially clarifying the value of the root by finding the values ​​of the next bits of the desired value of the root, starting with the highest one and moving to the lowest ones. For example, the value of the root at the first step turns out to be 2, at the second – 2.2, at the third – 2.23, and so on 2.236067977…. Let us describe how the values ​​of the digits are found.

The digits are found by searching through them possible values 0, 1, 2, …, 9. In this case, the nth powers of the corresponding numbers are calculated in parallel, and they are compared with the radical number. If at some stage the value of the degree exceeds the radical number, then the value of the digit corresponding to the previous value is considered found, and a transition is made to next step algorithm for extracting the root, but if this does not happen, then the value of this bit is 9.

Let us explain these points using the same example of extracting the square root of five.

First we find the value of the units digit. We will go through the values ​​0, 1, 2, ..., 9, calculating 0 2, 1 2, ..., 9 2, respectively, until we get a value greater than the radical number 5. It is convenient to present all these calculations in the form of a table:

So the value of the units digit is 2 (since 2 2<5 , а 2 3 >5). Let's move on to finding the value of the tenths place. In this case, we will square the numbers 2.0, 2.1, 2.2, ..., 2.9, comparing the resulting values ​​with the radical number 5:

Since 2.2 2<5 , а 2,3 2 >5, then the value of the tenths place is 2. You can proceed to finding the value of the hundredths place:

This is how the next value of the root of five was found, it is equal to 2.23. And so you can continue to find values: 2,236, 2,2360, 2,23606, 2,236067, … .

To consolidate the material, we will analyze the extraction of the root with an accuracy of hundredths using the considered algorithm.

First we determine the most significant digit. To do this, we cube the numbers 0, 10, 100, etc. until we get a number greater than 2,151,186. We have 0 3 =0<2 151,186 , 10 3 =1 000<2151,186 , 100 3 =1 000 000>2 151,186 , so the most significant digit is the tens digit.

Let's determine its value.

Since 10 3<2 151,186 , а 20 3 >2 151.186, then the value of the tens place is 1. Let's move on to units.

Thus, the value of the ones digit is 2. Let's move on to tenths.

Since even 12.9 3 is less than the radical number 2 151.186, then the value of the tenths place is 9. It remains to perform the last step of the algorithm; it will give us the value of the root with the required accuracy.

At this stage, the value of the root is found accurate to hundredths: .

In conclusion of this article, I would like to say that there are many other ways to extract roots. But for most tasks, the ones we studied above are sufficient.

References.

• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).

Instructions

To raise a number to the 1/3 power, enter the number, then click the exponentiation button and enter the approximate value of 1/3 - 0.333. This accuracy is quite sufficient for most calculations. However, the accuracy of calculations is very easy to increase - just add as many triplets as will fit on the calculator indicator (for example, 0.3333333333333333). Then click the "=" button.

To calculate the third root using a computer, run the Windows calculator program. The procedure for calculating the third root is completely similar to that described above. The only difference is in the design of the exponentiation button. On the virtual keyboard of the calculator it is indicated as “x^y”.

The third root can also be calculated in MS Excel. To do this, enter “=” in any cell and select the “insert” icon (fx). Select the “DEGREE” function in the window that appears and click the “OK” button. In the window that appears, enter the value of the number for which you want to calculate the third root. In "Degree" enter the number "1/3". Type the number 1/3 exactly in this form - like an ordinary one. After that, click the “Ok” button. The cube root of the given number will appear in the table cell where it was created.

If the third root has to be calculated constantly, then slightly improve the method described above. For the number from which you want to extract the root, indicate not the number itself, but a table cell. After that, just enter the original number into this cell each time - its cube root will appear in the cell with the formula.

Video on the topic

Please note

Conclusion. This paper examined various methods for calculating cube root values. It turned out that the values ​​of the cube root can be found using the iteration method, you can also approximate the cube root, raise the number to the power of 1/3, look for the values ​​of the third root using Microsoft Office Ecxel, setting formulas in cells.

Useful advice

Roots of the second and third degrees are used especially often and therefore have special names. Square root: In this case, the exponent is usually omitted, and the term "root" without specifying the exponent most often implies the square root. Practical calculation of roots Algorithm for finding the root of the nth degree. Square and cube roots are usually provided in all calculators.

Sources:

• third root
• How to take the square root to the Nth power in Excel

The operation of finding the root third degrees is usually called the extraction of the “cubic” root, and it consists in finding a real number, the cube of which will give a value equal to the radical number. The operation of extracting any arithmetic root degrees n is equivalent to the operation of raising to the power 1/n. There are several methods you can use to practically calculate the cube root.