The topic of square roots is mandatory in the school curriculum of mathematics. We can not 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, quite complex ones: exponentiation, multiplication and division. But there are simple enough: subtraction and addition of roots. By the way, they only seem so at first glance. Run them without errors is not always just for someone who is just starting to get to know them.
What is the mathematical root?
This action arose in opposition to the exponentiation. Mathematics assumes the existence of two opposite operations. There is a subtraction on addition. Multiplication is opposed by division. The inverse action of a degree is the extraction of the corresponding root.
If the degree is two, then the root will be square. It is the most common in school mathematics. He does not even have an indication that he is square, that is, there is not assigned the number 2. 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 from a certain number, it is necessary to find out what the radicand gives when multiplying by itself. This number and will be the square root. If we write this mathematically, we get the following: x * x = x 2 = y, then √y = x.
What actions can be taken with them?
At its core, the root is a fractional power, which has a unit in the numerator. A denominator can be any. For example, it has a square root of two. Therefore, all actions that can be performed with powers will also be valid for roots.
And the requirements for these actions they have the same. If multiplication, division and exponentiation do not encounter difficulties for students, then the addition of roots, as well as their subtraction, sometimes leads to confusion. And all because I want to perform these operations without looking at the root sign. And here mistakes begin.
What are the rules for adding and subtracting them?
First you need to remember two categorical "impossible":
- it is impossible to perform addition and subtraction of roots, as in primes, that is, it is impossible to write the sub-root expressions of a sum under one sign and perform mathematical operations with them;
- it is impossible to add and subtract roots with different indicators, for example square and cubic.
A clear example of the first prohibition:√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
- Find and group similar roots. That is, those who not only have the same numbers under the radical, but they themselves with the same indicator.
- Add the addition of roots, united in one group by the first action. It is easy to implement, because you only need to add up the values that stand before the radicals.
- Extract the roots in those terms in which the radicand 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.
- Extract from the sign of the radical multiplier, which gives the whole degree.
- See if there are any similar terms again. If yes, then perform the second action again.
In a situation where the task does not require an exact root value, it can be calculated on a calculator. Endless decimal fraction, which will be highlighted in its window, round off. Most often this is done up to the hundredth. And then perform all operations for decimal fractions.
Recommendation: after decomposition into prime factors, you need to make a check. That is, multiply them one by one and check whether the original value is obtained.
This is all information about how the addition of roots is performed. The examples below will illustrate the above.
The first task
Calculate the value of the expressions:
a) √2 + 3√32 + ½ √128 - 6√18;
a) If you follow the above algorithm, you can see that there is nothing for the first two steps in this example. But it is possible to simplify some subordinate expressions.
For example, 32 is decomposed into two factors of 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 sign of the radical those factors that give the square of the number. This is 16 = 4 2. 9 = 3 2. 64 = 8 2. The expression takes the form:
√2 + 3 * 4√2 + ½ * 8 √2 - 6 * 3√2.
You need to simplify the recording a bit. To do this, multiply the coefficients in front of the signs of the root:
√2 + 12√2 + 4 √2 - 12√2.
In this expression, all the terms turned out to be similar. So they just need to be folded. The answer is: 5√2.
b) Like the previous example, the addition of roots begins with their simplification. The subordinate expressions 75, 147, 48 and 300 will be represented by such pairs: 5 and 25, 3 and 49, 3 and 16, 3 and 100. In each of them there is a number that can be taken out from under the root sign:
After simplification, the answer is: 5√5 - 5√3. It can be left in this form, but it is better to take the common multiplier 5 for the bracket: 5 (√5 - √3).
Example with fractional expressions
To multiply it is necessary to expand such numbers: 45 = 5 * 9, 20 = 4 * 5, 18 = 2 * 9, 245 = 5 * 49. Analogously, we need to take the multipliers 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 that we get 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.
For completeness of action, it is necessary to isolate the whole part of the factors in front of the roots. At the first it is equal to 1, for the second - 2.