Actions with the roots of the formula addition and subtraction. How to add and subtract square roots

Properties square roots

So far, we have performed five arithmetic operations on numbers: addition, subtraction, multiplication, division and exponentiation, and various properties of these operations were actively used in calculations, for example, a + b = b + a, an-bn = (ab) n, etc.

This chapter introduces a new operation - taking the square root of a non-negative number. To successfully use it, you need to get acquainted with the properties of this operation, which we will do in this section.

Proof. Let us introduce the following notation: https://pandia.ru/text/78/290/images/image005_28.jpg" alt="(!LANG:Equality" width="120" height="25 id=">!}.

This is how we formulate the following theorem.

(A short formulation that is more convenient to use in practice: the root of a fraction is equal to the fraction of the roots, or the root of the quotient is equal to the quotient of the roots.)

This time we will give only a brief record of the proof, and you can try to make appropriate comments similar to those that made up the essence of the proof of Theorem 1.

Remark 3. Of course, this example can be solved differently, especially if you have a calculator at hand: multiply the numbers 36, 64, 9, and then take the square root of the resulting product. However, you will agree that the solution proposed above looks more cultural.

Remark 4. In the first method, we carried out head-on calculations. The second way is more elegant:
we applied formula a2 - b2 = (a - b) (a + b) and used the property of square roots.

Remark 5. Some "hotheads" sometimes offer the following "solution" to Example 3:

This, of course, is not true: you see - the result is not the same as in our example 3. The fact is that there is no property https://pandia.ru/text/78/290/images/image014_6.jpg" alt="(!LANG:Task" width="148" height="26 id=">!} There are only properties concerning the multiplication and division of square roots. Be careful and careful, do not take wishful thinking.

Concluding the paragraph, we note one more rather simple and at the same time important property:
if a > 0 and n - natural number , then

Converting Expressions Containing the Square Root Operation

So far, we have only performed transformations rational expressions, using for this the rules of operations on polynomials and algebraic fractions, formulas for abbreviated multiplication, etc. In this chapter, we introduced a new operation - the operation of extracting a square root; we have established that

where, recall, a, b are non-negative numbers.

Using these formulas, you can perform various transformations of expressions containing the square root operation. Let's consider several examples, and in all examples we will assume that the variables take only non-negative values.

Example 3 Enter a factor under the square root sign:

Example 6. Simplify the expression Solution. Let's perform successive transformations:

The topic of square roots is mandatory in the school curriculum of the mathematics course. You can't do without them when solving quadratic equations. And later it becomes necessary not only to extract the roots, but also to perform other actions with them. Among them are quite complex: exponentiation, multiplication and division. But there are also quite simple ones: subtraction and addition of roots. By the way, they only seem so at first glance. Performing them without errors is not always easy for someone who is just starting to get acquainted with them.

What is a mathematical root?

This action arose as opposed to exponentiation. Mathematics assumes the presence of two opposite operations. There is subtraction for addition. Multiplication is opposed to division. The reverse action of the degree is the extraction of the corresponding root.

If the exponent is 2, then the root will be square. It is the most common in school mathematics. It does not even have an indication that it is square, that is, the number 2 is not assigned to it. The mathematical notation of this operator (radical) is shown in the figure.

From the described action, its definition follows smoothly. To extract the square root of a certain number, you need to find out what the radical expression will give when multiplied by itself. This number will be the square root. If we write this mathematically, we get the following: x * x \u003d x 2 \u003d y, which means √y \u003d x.

What actions can be taken with them?

At its core, a root is a fractional power that has a unit in the numerator. And the denominator can be anything. For example, the square root has a value of two. Therefore, all actions that can be performed with degrees will also be valid for roots.

And they have the same requirements for these actions. If multiplication, division and exponentiation do not meet with difficulties for students, then the addition of roots, as well as their subtraction, sometimes leads to confusion. And all because you want to perform these operations without looking at the sign of the root. And this is where the mistakes begin.

What are the rules for addition and subtraction?

First you need to remember two categorical "no":

• it is impossible to perform addition and subtraction of roots, as with prime numbers, that is, it is impossible to write the root expressions of the sum under one sign and perform mathematical operations with them;
• you cannot add and subtract roots with different exponents, such as square and cubic.

An illustrative example of the first ban: √6 + √10 ≠ √16 but √(6 + 10) = √16.

In the second case, it is better to limit ourselves to simplifying the roots themselves. And in the answer leave their sum.

Now to the rules

1. Find and group similar roots. That is, those who not only have the same numbers under the radical, but they themselves have one indicator.
2. Perform the addition of the roots combined into one group by the first action. It is easy to implement, because you only need to add the values ​​that come before the radicals.
3. Extract the roots in those terms in which the radical expression forms a whole square. In other words, do not leave anything under the sign of the radical.
4. Simplify root expressions. To do this, you need to factor them into prime factors and see if they give the square of any number. It is clear that this is true when it comes to the square root. When the exponent is three or four, then the prime factors must give the cube or the fourth power of the number.
5. Take out from under the sign of the radical a factor that gives an integer power.
6. See if similar terms appear again. If yes, then perform the second step again.

In a situation where the problem does not require the exact value of the root, it can be calculated on a calculator. Round off the infinite decimal fraction that will be displayed in its window. Most often this is done up to the hundredths. And then perform all operations for decimal fractions.

This is all the information about how the addition of the roots is performed. The examples below will illustrate the above.

First task

Calculate the value of expressions:

a) √2 + 3√32 + ½ √128 - 6√18;

b) √75 - √147 + √48 - 1/5 √300;

c) √275 - 10√11 + 2√99 + √396.

a) If you follow the algorithm above, you can see that there is nothing for the first two actions in this example. But you can simplify some radical expressions.

For example, factor 32 into two factors 2 and 16; 18 will be equal to the product of 9 and 2; 128 is 2 by 64. Given this, the expression will be written like this:

√2 + 3√(2 * 16) + ½ √(2 * 64) - 6 √(2 * 9).

Now you need to take out from under the radical sign those factors that give the square of the number. This is 16=4 2 , 9=3 2 , 64=8 2 . The expression will take the form:

√2 + 3 * 4√2 + ½ * 8 √2 - 6 * 3√2.

We need to simplify the writing a bit. For this, the coefficients are multiplied before the signs of the root:

√2 + 12√2 + 4 √2 - 12√2.

In this expression, all the terms turned out to be similar. Therefore, they just need to be folded. The answer will be: 5√2.

b) Like the previous example, the addition of roots begins with their simplification. The root expressions 75, 147, 48 and 300 will be represented by the following pairs: 5 and 25, 3 and 49, 3 and 16, 3 and 100. Each of them has a number that can be taken out from under the root sign:

5√5 - 7√3 + 4√3 - 1/5 * 10√3.

After simplification, the answer is: 5√5 - 5√3. It can be left in this form, but it is better to take the common factor 5 out of the bracket: 5 (√5 - √3).

c) And again factorization: 275 = 11 * 25, 99 = 11 * 9, 396 = 11 * 36. After factoring out the root sign, we have:

5√11 - 10√11 + 2 * 3√11 + 6√11. After reducing similar terms, we get the result: 7√11.

Fractional example

√(45/4) - √20 - 5√(1/18) - 1/6 √245 + √(49/2).

The following numbers will need to be factored: 45 = 5 * 9, 20 = 4 * 5, 18 = 2 * 9, 245 = 5 * 49. Similarly to those already considered, you need to take the factors out from under the root sign and simplify the expression:

3/2 √5 - 2√5 - 5/ 3 √(½) - 7/6 √5 + 7 √(½) = (3/2 - 2 - 7/6) √5 - (5/3 - 7 ) √(½) = - 5/3 √5 + 16/3 √(½).

This expression requires getting rid of the irrationality in the denominator. To do this, multiply the second term by √2/√2:

5/3 √5 + 16/3 √(½) * √2/√2 = - 5/3 √5 + 8/3 √2.

To complete the action, you need to select the integer part of the factors in front of the roots. The first is 1, the second is 2.

The root of a number is easiest to subtract using a calculator. But, if you do not have a calculator, then you need to know the algorithm for calculating the square root. The fact is that a number in a square sits under the root. For example, 4 squared is 16. That is, the square root of 16 will be equal to four. Also, 5 squared is 25. Therefore, the root of 25 will be 5. And so on.

If the number is small, then it can be easily subtracted verbally, for example, the root of 25 will be 5, and the root of 144-12. You can also calculate on the calculator, there is a special root icon, you need to drive in a number and click on the icon.

The square root table will also help:

There are other ways that are more complex, but very effective:

The root of any number can be subtracted using a calculator, especially since they are in every phone today.

You can try to figure out how it might turn out given number by multiplying one number by itself.



Calculating the square root of a number is not difficult, especially if there is a special table. A well-known table from algebra lessons. Such an operation is called extracting the square root of the number aquot ;, in other words, solving the equation. Almost all calculators in smartphones have a square root function.



The result of extracting the square root of a known number will be another number, which, when raised to the second power (square), will give the same number that we know. Consider one of the descriptions of the settlements, which seems short and understandable:







Here is a video on the topic:

There are several ways to calculate the square root of a number.

The most popular way is to use a special root table (see below).

Also on each calculator there is a function with which you can find the root.

Or using a special formula.



There are several ways to extract the square root of a number. One of them is the fastest, using a calculator.

But if there is no calculator, then you can do it manually.

The result will be accurate.

The principle is almost the same as division by a column:



















Let's try without a calculator to find the value of the square root of a number, for example, 190969.







Thus, everything is extremely simple. In calculations, the main thing is to adhere to certain simple rules and think logically.

For this you need a table of squares



For example, the root of 100 = 10, of 20 = 400 of 43 = 1849

Now almost all calculators, including those on smartphones, can calculate the square root of a number. BUT if you don’t have a calculator, then you can find the root of the number in several simple ways:

Prime factorization



Factor the root number into factors that are square numbers. Depending on the root number, you will get an approximate or exact answer. Square numbers are numbers from which the whole square root can be taken. Factors of a number that, when multiplied, give the original number. For example, the factors of the number 8 are 2 and 4, since 2 x 4 = 8, the numbers 25, 36, 49 are square numbers, since 25 = 5, 36 = 6, 49 = 7. Square factors are factors that are square numbers . First, try to factorize the root number into square factors.

For example, calculate the square root of 400 (manually). First try factoring 400 into square factors. 400 is a multiple of 100, which is a square number divisible by 25. Dividing 400 by 25 gives you 16, which is also a square number. Thus, 400 can be factored into square factors of 25 and 16, that is, 25 x 16 = 400.

Write it down as: 400 = (25 x 16).



The square root of the product of some terms is equal to the product of the square roots of each term, that is, (a x b) = a x b. Using this rule, take the square root of each square factor and multiply the results to find the answer.

In our example, take the square root of 25 and 16.



If the root number does not factor into two square factors (and it does in most cases), you will not be able to find the exact answer as a whole number. But you can simplify the problem by decomposing the root number into a square factor and an ordinary factor (a number from which the whole square root cannot be taken). Then you will take the square root of the square factor and you will take the root of the ordinary factor.

For example, calculate the square root of the number 147. The number 147 cannot be factored into two square factors, but it can be factored into the following factors: 49 and 3. Solve the problem as follows:



Now you can evaluate the value of the root (find an approximate value) by comparing it with the values ​​of the square roots that are closest (on both sides of the number line) to the root number. You will get the value of the root as decimal fraction, which must be multiplied by the number behind the root sign.

Let's go back to our example. The root number is 3. The nearest square numbers to it will be the numbers 1 (1 \u003d 1) and 4 (4 \u003d 2). Thus, the value of 3 is between 1 and 2. Since the value of 3 is probably closer to 2 than to 1, our estimate is: 3 = 1.7. We multiply this value by the number at the root sign: 7 x 1.7 \u003d 11.9. If you do the calculations on a calculator, you get 12.13, which is pretty close to our answer.

This method also works with large numbers. For example, consider 35. The root number is 35. The nearest square numbers to it are 25 (25 = 5) and 36 (36 = 6). Thus, the value 35 is between 5 and 6. Since the value 35 is much closer to 6 than to 5 (because 35 is only 1 less than 36), we can say that 35 is slightly less than 6. Checking on the calculator gives us the answer 5.92 - we were right.



Another way is to factorize the root number into prime factors. Prime factors of a number that are only divisible by 1 and themselves. Write the prime factors in a row and find pairs of identical factors. Such factors can be taken out of the sign of the root.

For example, calculate the square root of 45. We decompose the root number into prime factors: 45 \u003d 9 x 5, and 9 \u003d 3 x 3. Thus, 45 \u003d (3 x 3 x 5). 3 can be taken out of the root sign: 45 = 35. Now we can estimate 5.

Consider another example: 88.

= (2 x 4 x 11)

= (2 x 2 x 2 x 11). You got three multiplier 2s; take a couple of them and take them out of the sign of the root.

2(2 x 11) = 22 x 11. Now you can evaluate 2 and 11 and find an approximate answer.

This tutorial video may also be helpful:

To extract the root from a number, you should use a calculator, or if there is no suitable one, I advise you to go to this site and solve the problem using online calculator, which will give the correct value in seconds.

Content:

Adding and subtracting square roots is possible only if they have the same root expression, that is, you can add or subtract 2√3 and 4√3, but not 2√3 and 2√5. You can simplify the root expression to convert them to roots with the same radical expression (and then add or subtract them).

Steps

Part 1 Understanding the Basics

1. 1 (expression under the sign of the root). To do this, decompose the root number into two factors, one of which is a square number (a number from which a whole root can be extracted, for example, 25 or 9). After that, take the root of the square number and write down the found value in front of the root sign (the second factor will remain under the root sign). For example, 6√50 - 2√8 + 5√12. The numbers in front of the root sign are the factors of the corresponding roots, and the numbers under the root sign are the radical numbers (expressions). Here's how to solve this problem:
   • 6√50 = 6√(25 x 2) = (6 x 5)√2 = 30√2. Here you factor 50 into factors 25 and 2; then from 25 you extract the root equal to 5, and take out 5 from under the root. Then multiply 5 by 6 (factor at the root) and get 30√2.
   • 2√8 = 2√(4 x 2) = (2 x 2)√2 = 4√2. Here you factor 8 into factors 4 and 2; then from 4 you extract the root equal to 2, and take 2 out from under the root. Then you multiply 2 by 2 (factor at the root) and you get 4√2.
   • 5√12 = 5√(4 x 3) = (5 x 2)√3 = 10√3. Here you factor 12 into factors 4 and 3; then from 4 you extract the root equal to 2, and take 2 out from under the root. Then you multiply 2 by 5 (factor at the root) and you get 10√3.
2. 2 Underline the roots whose root expressions are the same. In our example, the simplified expression is: 30√2 - 4√2 + 10√3. In it, you must underline the first and second terms ( 30√2 and 4√2 ), since they have the same root number 2. Only such roots can you add and subtract.
3. 3 If you are given an expression with a large number of terms, many of which have the same radical expressions, use single, double, triple underscores to indicate such terms to make it easier to solve this expression.
4. 4 At roots whose radical expressions are the same, add or subtract the factors in front of the root sign, and leave the radical expression the same (do not add or subtract radical numbers!). The idea is to show how many roots with a certain radical expression are contained in this expression.
   • 30√2 - 4√2 + 10√3 =
   • (30 - 4)√2 + 10√3 =
   • 26√2 + 10√3

Part 2 Practicing with examples

1. 1 Example 1: √(45) + 4√5.
   • Simplify √(45). Factor 45: √(45) = √(9 x 5).
   • Move 3 out from under the root (√9 = 3): √(45) = 3√5.
   • Now add the factors at the roots: 3√5 + 4√5 = 7√5
2. 2 Example 2: 6√(40) - 3√(10) + √5.
   • Simplify 6√(40). Factor 40: 6√(40) = 6√(4 x 10).
   • Move 2 out from under the root (√4 = 2): 6√(40) = 6√(4 x 10) = (6 x 2)√10.
   • Multiply the factors before the root and get 12√10.
   • Now the expression can be written as 12√10 - 3√(10) + √5. Since the first two terms have the same radical numbers, you can subtract the second term from the first, and leave the first unchanged.
   • You will get: (12-3)√10 + √5 = 9√10 + √5.
3. 3 Example 3 9√5 -2√3 - 4√5. Here, none of the radical expressions can be factorized, so simplifying this expression will not work. You can subtract the third term from the first (since they have the same root number) and leave the second term unchanged. You will get: (9-4)√5 -2√3 = 5√5 - 2√3.
4. 4 Example 4 √9 + √4 - 3√2.
   • √9 = √(3 x 3) = 3.
   • √4 = √(2 x 2) = 2.
   • Now you can just add 3 + 2 to get 5.
   • Final answer: 5 - 3√2.
5. 5 Example 5 Solve an expression containing roots and fractions. You can only add and calculate fractions that have a common (same) denominator. The expression (√2)/4 + (√2)/2 is given.
   • Find the smallest common denominator of these fractions. This is a number that is evenly divisible by each denominator. In our example, the number 4 is divisible by 4 and 2.
   • Now multiply the second fraction by 2/2 (to bring it to a common denominator; the first fraction has already been reduced to it): (√2)/2 x 2/2 = (2√2)/4.
   • Add up the numerators and leave the denominator the same: (√2)/4 + (2√2)/4 = (3√2)/4 .

• Before adding or subtracting roots, be sure to simplify (if possible) the radical expressions.

Warnings

• Never add or subtract roots with different root expressions.
• Never add or subtract an integer and a root, for example, 3 + (2x) 1/2 .
  • Note: "x" to the second power and the square root of "x" are the same thing (i.e. x 1/2 = √x).

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