The topic of square roots is mandatory in the school curriculum of the mathematics course. Without them, not to do 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 quite simple: subtraction and addition of roots. By the way, they only seem so at first glance. Running them without errors is not always easy for someone who is just starting to get acquainted with them.
What is the math root?
This action originated in opposition to exponentiation. Mathematics assumes the existence of two opposite operations. There is subtraction for addition. Multiplication is opposed by division. The inverse of the degree is the extraction of the corresponding root.
If there is a two in the power, then the root will be square. It is most common in school math. He does not even have an indication that he is square, that is, the figure 2 is not attributed to him. The mathematical notation of this operator (radical) is shown in the figure.
From the described action smoothly follows his definition. To extract the square root of a number, you need to figure out which one will give when you multiply the self-contained expression. This number will be the square root. If you write it mathematically, you get the following: x * x = x 2 = y, then √у = x.
What actions can be performed with them?
At its core, the root is a fractional degree, in which there is a unit in the numerator. And the denominator can be any. For example, at the square root, it is equal to two. Therefore, all actions that can be performed with degrees will be valid for roots.
And the requirements for these actions are the same. If multiplication, division, and exponentiation do not encounter difficulties for students, the addition of roots, like their subtraction, sometimes leads to confusion. And all because I want to perform these operations without regard to the root sign. And here the mistakes begin.
What are the rules for adding and subtracting them?
First you need to remember two categorical "no":
- you cannot perform addition and subtraction of roots, as in primes, that is, it is impossible to write single-sign root mean-sum expressions and perform mathematical operations with them;
- you can not add and subtract roots with different indicators, such as square and cubic.
A good 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 to leave their amount.
Now to the rules
- Find and group similar roots. That is, those who have not only the same numbers under the radical, but they themselves have the same indicator.
- To perform the addition of the roots, united in one group by the first action. It is easy to implement, because you just need to add up the values that stand in front of the radicals.
- 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.
- To simplify radical expressions. For this you need to decompose them into Prime factors and see if they'll give square any number. It is clear that this is true if we are talking about square root. When the exponent three or four, then the Prime factors should give the cube or fourth power of the number.
- Remove from under the sign of the radical factor, which gives the whole degree.
- See if similar terms appear again. If so, then perform the second action again.
In a situation where the task does not require the exact value of the root, it can be calculated on the calculator. Endless decimal fraction, which will be displayed in its window, round. Most often it is done to the hundredth. And then perform all operations for decimal fractions.
Recommendation: after decomposition into prime factors, you need to check. That is, multiply them on each other and check whether the original value is obtained.
This is all information on how the addition of roots is performed. The examples below will illustrate the above.
Calculate the value of expressions:
a) √2 + 3√32 + ½ √128 - 6√18;
a) If you follow the above algorithm, it is clear that for the first two actions in this example there is nothing. But you can simplify some radical expressions.
For example, 32 is decomposed 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 we need to remove from the radical sign those factors that give the square of a number. This 16 = 4 2. 9 = 3 2. 64 = 8 2. The expression will look like:
√2 + 3 * 4√2 + ½ * 8 √2 - 6 * 3√2.
Need a little easier to write. To do this, multiply the coefficients in front of the root signs:
√2 + 12√2 + 4 √2 - 12√2.
In this expression, all the terms turned out to be similar. Therefore, they just need to fold. The answer will be: 5√2.
b) Like the previous example, the addition of the roots begins with their simplification. The radical 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 removed from the root sign:
After simplification, we get the answer: 5√5 - 5√3. It can be left in this form, but it is better to take the common factor 5 for the bracket: 5 (√5 - √3).
Example with fractional expressions
The factors will need to decompose the following numbers: 45 = 5 * 9, 20 = 4 * 5, 18 = 2 * 9, 245 = 5 * 49. Similarly to those already considered, you need to remove the factors from 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 to get rid of irrationality in the denominator. To do this, multiply by √2 / √2 the second term:
- 5/3 √5 + 16/3 √ (½) * √2 / √2 = - 5/3 √5 + 8/3 √2.
For completeness of action, it is necessary to select the integer part of the factors before the roots. In the first, it is equal to 1, in the second - 2.