Is it possible to add roots? How to add square roots

Root formulas. Properties of square roots.

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In the previous lesson we figured out what a square root is. It's time to figure out which ones exist formulas for roots what are properties of roots, and what can be done with all this.

Formulas of roots, properties of roots and rules for working with roots- this is essentially the same thing. Formulas for square roots surprisingly little. Which certainly makes me happy! Or rather, you can write a lot of different formulas, but for practical and confident work with roots, only three are enough. Everything else flows from these three. Although many people get confused in the three root formulas, yes...

Let's start with the simplest one. Here she is:

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You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

In mathematics, any action has its opposite pair - in essence, this is one of the manifestations of the Hegelian law of dialectics: “the unity and struggle of opposites.” One of the actions in such a “pair” is aimed at increasing the number, and the other, its opposite, is aimed at decreasing it. For example, the opposite of addition is subtraction, and division is the opposite of multiplication. Exponentiation also has its own dialectical opposite pair. We are talking about extracting the root.

To extract the root of such and such a power from a number means to calculate which number must be raised to the appropriate power in order to end up with a given number. The two degrees have their own separate names: the second degree is called “square”, and the third is called “cube”. Accordingly, it is nice to call the roots of these powers square and cubic roots. Actions with cube roots are a topic for a separate discussion, but now let's talk about adding square roots.

Let's start with the fact that in some cases it is easier to first extract square roots and then add the results. Suppose we need to find the value of the following expression:

After all, it’s not at all difficult to calculate that the square root of 16 is 4, and of 121 is 11. Therefore,

√16+√121=4+11=15

However, this is the simplest case - here we're talking about about perfect squares, i.e. about those numbers that are obtained by squaring integers. But this doesn't always happen. For example, the number 24 is not a perfect square (there is no integer that, when raised to the second power, would result in 24). The same applies to a number like 54... What if we need to add the square roots of these numbers?

In this case, we will receive in the answer not a number, but another expression. The maximum we can do here is to simplify the original expression as much as possible. To do this, you will have to take out the factors from under the square root. Let's see how this is done using the numbers mentioned above as an example:

To begin with, let's factor 24 into factors so that one of them can easily be extracted as a square root (i.e., so that it is a perfect square). There is such a number – it’s 4:

Now let's do the same with 54. In its composition, this number will be 9:

Thus, we get the following:

√24+√54=√(4*6)+ √(9*6)

Now let’s extract the roots from what we can extract them from: 2*√6+3*√6

There is a common factor here that we can take out of brackets:

(2+3)* √6=5*√6

This will be the result of addition - nothing more can be extracted here.

True, you can resort to using a calculator - however, the result will be approximate and with a huge number of decimal places:

√6=2,449489742783178

Gradually rounding it up, we get approximately 2.5. If we would still like to bring it to logical conclusion solution to the previous example, we can multiply this result by 5 - and we get 12.5. It is impossible to obtain a more accurate result with such initial data.

Square root of a number X called number A, which in the process of multiplying by itself ( A*A) can give a number X.
Those. A * A = A 2 = X, And √X = A.

Above square roots ( √x), like other numbers, you can perform arithmetic operations such as subtraction and addition. To subtract and add roots, they need to be connected using signs corresponding to these actions (for example √x - √y ).
And then bring the roots to their simplest form - if there are similar ones between them, it is necessary to make a reduction. It consists in taking the coefficients of similar terms with the signs of the corresponding terms, then putting them in brackets and deducing the common root outside the brackets of the factor. The coefficient we obtained is simplified according to the usual rules.

Step 1: Extracting square roots

Firstly, to add square roots, you first need to extract these roots. This can be done if the numbers under the root sign are perfect squares. For example, take the given expression √4 + √9 . First number 4 is the square of the number 2 . Second number 9 is the square of the number 3 . Thus, we can obtain the following equality: √4 + √9 = 2 + 3 = 5 .
That's it, the example is solved. But it doesn’t always happen that easily.

Step 2. Taking out the multiplier of the number from under the root

If there are no perfect squares under the root sign, you can try to remove the multiplier of the number from under the root sign. For example, let's take the expression √24 + √54 .

Factor the numbers:
24 = 2 * 2 * 2 * 3 ,
54 = 2 * 3 * 3 * 3 .

Among 24 we have a multiplier 4 , it can be taken out from under the square root sign. Among 54 we have a multiplier 9 .

We get equality:
√24 + √54 = √(4 * 6) + √(9 * 6) = 2 * √6 + 3 * √6 = 5 * √6 .

Considering this example, we obtain the removal of the multiplier from under the root sign, thereby simplifying the given expression.

Step 3: Reducing the Denominator

Consider the following situation: the sum of two square roots is the denominator of the fraction, for example, A/(√a + √b).
Now we are faced with the task of “getting rid of irrationality in the denominator.”
Let's use the following method: multiply the numerator and denominator of the fraction by the expression √a - √b.

We now get the abbreviated multiplication formula in the denominator:
(√a + √b) * (√a - √b) = a - b.

Similarly, if the denominator has a root difference: √a - √b, the numerator and denominator of the fraction are multiplied by the expression √a + √b.

Let's take the fraction as an example:
4 / (√3 + √5) = 4 * (√3 - √5) / ((√3 + √5) * (√3 - √5)) = 4 * (√3 - √5) / (-2) = 2 * (√5 - √3) .

Example of complex denominator reduction

Now we will consider a rather complex example of getting rid of irrationality in the denominator.

For example, let's take a fraction: 12 / (√2 + √3 + √5) .
You need to take its numerator and denominator and multiply by the expression √2 + √3 - √5 .

We get:

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

Step 4. Calculate the approximate value on the calculator

If you only need an approximate value, this can be done on a calculator by calculating the value of the square roots. The value is calculated separately for each number and written down with the required accuracy, which is determined by the number of decimal places. Next, all the required operations are performed, as with ordinary numbers.

Example of calculating an approximate value

It is necessary to calculate the approximate value of this expression √7 + √5 .

As a result we get:

√7 + √5 ≈ 2,65 + 2,24 = 4,89 .

Please note: under no circumstances should you add square roots as prime numbers; this is completely unacceptable. That is, if we add the square root of five and the square root of three, we cannot get the square root of eight.

Helpful advice: if you decide to factor a number, in order to derive the square from under the root sign, you need to do a reverse check, that is, multiply all the factors that resulted from the calculations, and the final result of this mathematical calculation should be the number that was originally given to us.

In our time with modern electronic computers, calculating the root of a number is not possible challenging task. For example, √2704=52, any calculator will calculate this for you. Fortunately, the calculator is available not only in Windows, but also in an ordinary, even the simplest, phone. True, if suddenly (with a small degree of probability, the calculation of which, by the way, includes adding the roots) you find yourself without available funds, then, alas, you will have to rely only on your brains.

Mind training never fails. Especially for those who don’t work with numbers that often, much less with roots. Adding and subtracting roots is a good workout for a bored mind. I will also show you how to add roots step by step. Examples of expressions may be as follows.

Equation to simplify:

√2+3√48-4×√27+√128

This is an irrational expression. In order to simplify it, you need to reduce all radical expressions to general appearance. We do it step by step:

The first number can no longer be simplified. Let's move on to the second term.

3√48 we factor 48: 48=2×24 or 48=3×16. of 24 is not an integer, i.e. has a fractional remainder. Since we need an exact value, approximate roots are not suitable for us. The square root of 16 is 4, take it out from under We get: 3×4×√3=12×√3

Our next expression is negative, i.e. written with a minus sign -4×√(27.) We factor 27. We get 27=3×9. We do not use fractional factors because it is more difficult to calculate the square root of fractions. We take 9 out from under the sign, i.e. calculate the square root. We get the following expression: -4×3×√3 = -12×√3

The next term √128 calculates the part that can be taken out from under the root. 128=64×2, where √64=8. If it makes it easier for you, you can imagine this expression like this: √128=√(8^2×2)

We rewrite the expression with simplified terms:

√2+12×√3-12×√3+8×√2

Now we add the numbers using the same radical expression. You cannot add or subtract expressions with different radical expressions. Adding roots requires compliance with this rule.

We get the following answer:

√2+12√3-12√3+8√2=9√2

√2=1×√2 - I hope the fact that in algebra it is customary to omit such elements will not be news to you.

Expressions can be represented not only by the square root, but also by the cubic or nth root.

Adding and subtracting roots with different indicators degrees, but with an equivalent radical expression, occurs as follows:

If we have an expression of the form √a+∛b+∜b, then we can simplify this expression as follows:

∛b+∜b=12×√b4 +12×√b3

12√b4 +12×√b3=12×√b4 + b3

We brought two similar members to overall indicator root Here the property of roots was used, which states: if the number of the degree of the radical expression and the number of the exponent of the root are multiplied by the same number, then its calculation will remain unchanged.

Note: exponents only add when multiplying.

Let's consider an example when the expression contains fractions.

5√8-4×√(1/4)+√72-4×√2

We will decide in stages:

5√8=5*2√2 - we take out the extracted part from under the root.

4√(1/4)=-4 √1/(√4)= - 4 *1/2= - 2

If the body of the root is represented by a fraction, then often this fraction will not change if you take the square root of the dividend and divisor. As a result, we received the equality described above.

√72-4√2=√(36×2)- 4√2=2√2

10√2+2√2-2=12√2-2

Here is the answer.

The main thing to remember is that a root with an even exponent cannot be extracted from negative numbers. If the radical expression of even degree is negative, then the expression is unsolvable.

Addition of roots is possible only if the radical expressions coincide, since they are similar terms. The same applies to the difference.

The addition of roots with different numerical exponents is carried out by reducing both terms to a common root degree. This law works the same way as reduction to a common denominator when adding or subtracting fractions.

If a radical expression contains a number raised to a power, then this expression can be simplified provided that there is a common denominator between the exponent of the root and the power.

Extracting the quadrant root of a number is not the only operation that can be performed with this mathematical phenomenon. Just like regular numbers, square roots add and subtract.

Rules for adding and subtracting square roots

Definition 1

Operations such as addition and subtraction of square roots are only possible if the radical expression is the same.

Example 1

You can add or subtract expressions 2 3 and 6 3, but not 5 6 And 9 4. If it is possible to simplify the expression and reduce it to roots with the same radical, then simplify and then add or subtract.

Actions with roots: basics

Example 2

6 50 - 2 8 + 5 12

Action algorithm:

Simplify the radical expression. To do this, it is necessary to decompose the radical expression into 2 factors, one of which is a square number (the number from which the whole square root is extracted, for example, 25 or 9).
Then you need to take the root of the square number and write the resulting value before the root sign. Please note that the second factor is entered under the sign of the root.
After the simplification process, it is necessary to emphasize the roots with the same radical expressions - only they can be added and subtracted.
For roots with the same radical expressions, it is necessary to add or subtract the factors that appear before the root sign. The radical expression remains unchanged. You cannot add or subtract radical numbers!

Tip 1

If you have an example with big amount identical radical expressions, then underline such expressions with single, double and triple lines to facilitate the calculation process.

Example 3

Let's try to solve this example:

6 50 = 6 (25 × 2) = (6 × 5) 2 = 30 2. First you need to decompose 50 into 2 factors 25 and 2, then take the root of 25, which is equal to 5, and take 5 out from under the root. After this, you need to multiply 5 by 6 (the factor at the root) and get 30 2.

2 8 = 2 (4 × 2) = (2 × 2) 2 = 4 2. First you need to decompose 8 into 2 factors: 4 and 2. Then take the root from 4, which is equal to 2, and take 2 out from under the root. After this, you need to multiply 2 by 2 (the factor at the root) and get 4 2.

5 12 = 5 (4 × 3) = (5 × 2) 3 = 10 3. First you need to decompose 12 into 2 factors: 4 and 3. Then extract the root of 4, which is equal to 2, and remove it from under the root. After this, you need to multiply 2 by 5 (the factor at the root) and get 10 3.

Simplification result: 30 2 - 4 2 + 10 3

30 2 - 4 2 + 10 3 = (30 - 4) 2 + 10 3 = 26 2 + 10 3 .

As a result, we saw how many identical radical expressions are contained in in this example. Now let's practice with other examples.

Example 4

Let's simplify (45). Factor 45: (45) = (9 × 5) ;
We take 3 out from under the root (9 = 3): 45 = 3 5 ;
Add the factors at the roots: 3 5 + 4 5 = 7 5.

Example 5

6 40 - 3 10 + 5:

Let's simplify 6 40. We factor 40: 6 40 = 6 (4 × 10) ;
We take 2 out from under the root (4 = 2): 6 40 = 6 (4 × 10) = (6 × 2) 10 ;
We multiply the factors that appear in front of the root: 12 10 ;
We write the expression in a simplified form: 12 10 - 3 10 + 5 ;
Since the first two terms have the same radical numbers, we can subtract them: (12 - 3) 10 = 9 10 + 5.

Example 6

As we can see, it is not possible to simplify radical numbers, so we look for terms with the same radical numbers in the example, carry out mathematical operations (add, subtract, etc.) and write the result:

(9 - 4) 5 - 2 3 = 5 5 - 2 3 .

Adviсe:

Before adding or subtracting, it is necessary to simplify (if possible) the radical expressions.
Adding and subtracting roots with different radical expressions is strictly prohibited.
You should not add or subtract a whole number or root: 3 + (2 x) 1 / 2 .
When performing operations with fractions, you need to find a number that is divisible by each denominator, then bring the fractions to a common denominator, then add the numerators, and leave the denominators unchanged.

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