The topic about square roots is mandatory in school curriculum mathematics course. You cannot 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 like that 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 in opposition to exponentiation. Mathematics suggests two opposing operations. There is subtraction for addition. Multiplication is opposed to division. The reverse 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. It does not even have an indication that it is square, that is, the number 2 is not assigned next to it. The mathematical notation of this operator (radical) is presented in the figure.
Its definition flows smoothly from the action described. To extract the square root of a 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 down mathematically, we get the following: x*x=x 2 =y, which means √y=x.
What actions can you perform with them?
At its core, a root is a fractional power with one in the numerator. And the denominator can be anything. For example, at square root it is equal to two. Therefore, all actions that can be performed with powers will also be valid for roots.
And the requirements for these actions are the same. If multiplication, division and exponentiation do not encounter difficulties for students, then adding roots, like subtracting them, sometimes leads to confusion. And all because I want to perform these operations without regard to the sign of the root. And this is where the mistakes begin.
What are the rules for adding and subtracting?
First you need to remember two categorical “don’ts”:
- it is impossible to perform addition and subtraction of roots, as with prime numbers, that is, it is impossible to write radical expressions of the sum under one sign and perform mathematical operations with them;
- You cannot add and subtract roots with different exponents, 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 leave their amount in the answer.
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 have the same indicator.
- Perform the addition of the roots combined into one group in the first action. It is easy to implement because you only need to add the values that appear in front of the radicals.
- Extract the roots of those terms in which the radical expression forms a whole square. In other words, do not leave anything under the sign of a radical.
- Simplify radical expressions. To do this, you need to decompose them into prime factors and see if they give the square of any number. It is clear that this is true when we are talking about 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.
- Remove from under the sign of the radical the factor that gives the whole power.
- See if similar terms appear again. If yes, then perform the second step again.
In a situation where the task does not require exact value root, it can be calculated on a calculator. Endless decimal, which will appear in its window, round up. Most often this is done to hundredths. And then perform all operations for decimal fractions.
This is all the information about how to add roots. 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 above algorithm, you can see that there is nothing for the first two actions in this example. But you can simplify some radical expressions.
For example, decompose 32 into two factors 2 and 16; 18 will be equal to the product of 9 and 2; 128 is 2 over 64. Given this, the expression will be written like this:
√2 + 3√(2 * 16) + ½ √(2 * 64) - 6 √(2 * 9).
Now you need to remove 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 recording a little. To do this, multiply the coefficients before the root signs:
√2 + 12√2 + 4 √2 - 12√2.
In this expression, all terms turned out to be similar. Therefore, you just need to fold them. The answer will be: 5√2.
b) Similar to the previous example, adding roots begins with simplifying them. The radical expressions 75, 147, 48 and 300 will be represented in the following pairs: 5 and 25, 3 and 49, 3 and 16, 3 and 100. Each of them contains 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 brackets: 5 (√5 - √3).
c) And again factorization: 275 = 11 * 25, 99 = 11 * 9, 396 = 11 * 36. After removing the factors from under the root sign, we have:
5√11 - 10√11 + 2 * 3√11 + 6√11. After bringing similar terms we get the result: 7√11.
Example with fractional expressions
√(45/4) - √20 - 5√(1/18) - 1/6 √245 + √(49/2).
You will need to factor the following numbers: 45 = 5 * 9, 20 = 4 * 5, 18 = 2 * 9, 245 = 5 * 49. Similar to those already discussed, you need to remove the factors 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 irrationality in the denominator. To do this, you need to multiply the second term by √2/√2:
5/3 √5 + 16/3 √(½) * √2/√2 = - 5/3 √5 + 8/3 √2.
To complete the actions, you need to select the whole part of the factors in front of the roots. For the first it is 1, for the second it is 2.
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, for addition square roots First you 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 full squares no 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 a 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 let's consider enough complex example 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 like 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.
The square root of a number x is a number a, which when multiplied by itself gives the number x: a * a = a^2 = x, ?x = a. As with any numbers, you can perform arithmetic operations of addition and subtraction with square roots.
Instructions
1. First, when adding square roots, try to extract these roots. This will be acceptable if the numbers under the root sign are perfect squares. Let's say the given expression is ?4 + ?9. The first number 4 is the square of the number 2. The second number 9 is the square of the number 3. Thus it turns out that: ?4 + ?9 = 2 + 3 = 5.
2. If there are no complete squares under the root sign, then try moving the multiplier of the number from under the root sign. Let's say, let's say the expression is given?24 +?54. Factor the numbers: 24 = 2 * 2 * 2 * 3, 54 = 2 * 3 * 3 * 3. The number 24 has a factor of 4, the one that can be transferred from under the square root sign. The number 54 has a factor of 9. Thus, it turns out that: ?24 + ?54 = ?(4 * 6) + ?(9 * 6) = 2 * ?6 + 3 * ?6 = 5 * ?6. IN in this example As a result, removing the factor from under the root sign resulted in simplifying the given expression.
3. Let the sum of 2 square roots be the denominator of a fraction, say A / (?a + ?b). And let your task be “to get rid of irrationality in the denominator.” Then you can use the next method. Multiply the numerator and denominator of the fraction by the expression ?a - ?b. Thus, the denominator will contain the abbreviated multiplication formula: (?a + ?b) * (?a - ?b) = a - b. By analogy, if the difference between the roots is given in the denominator: ?a - ?b, then the numerator and denominator of the fraction must be multiplied by the expression ?a + ?b. For example, let the fraction 4 / (?3 + ?5) = 4 * (?3 - ?5) / ((?3 + ?5) * (?3 - ?5)) = 4 * (?3 - ?5) / (-2) = 2 * (?5 - ?3).
4. Consider a more complex example of getting rid of irrationality in the denominator. Let the fraction 12 / (?2 + ?3 + ?5) be given. You need to multiply the numerator and denominator of the fraction by the expression?2 + ?3 - ?5:12 / (?2 + ?3 + ?5) = 12 * (?2 + ?3 - ?5) / ((?2 + ?3 + ?5) * (?2 + ?3 - ?5)) = 12 * (?2 + ?3 - ?5) / (2 * ?6) = ?6 * (?2 + ?3 - ?5) = 2 * ?3 + 3 * ?2 - ?30.
5. And finally, if you only need an approximate value, you can calculate the square roots using a calculator. Calculate the values separately for the entire number and write it down to the required precision (say, two decimal places). And after that, perform the required arithmetic operations, as with ordinary numbers. Let's say, let's say you need to find out the approximate value of the expression ?7 + ?5 ? 2.65 + 2.24 = 4.89.
Video on the topic
Pay attention!
In no case can square roots be added as primitive numbers, i.e. ?3 + ?2 ? ?5!!!
Useful advice
If you are factoring a number in order to move the square from under the root sign, then perform the reverse check - multiply all the resulting factors and get the original number.
Fact 1.
\(\bullet\) Let's take some non-negative number \(a\) (that is, \(a\geqslant 0\) ). Then (arithmetic) square root from the number \(a\) is called such a non-negative number \(b\) , when squared we get the number \(a\) : \[\sqrt a=b\quad \text(same as )\quad a=b^2\] From the definition it follows that \(a\geqslant 0, b\geqslant 0\). These restrictions are an important condition the existence of a square root and they should be remembered!
Remember that any number when squared gives a non-negative result. That is, \(100^2=10000\geqslant 0\) and \((-100)^2=10000\geqslant 0\) .
\(\bullet\) What is \(\sqrt(25)\) equal to? We know that \(5^2=25\) and \((-5)^2=25\) . Since by definition we must find a non-negative number, then \(-5\) is not suitable, therefore, \(\sqrt(25)=5\) (since \(25=5^2\) ).
Finding the value of \(\sqrt a\) is called taking the square root of the number \(a\) , and the number \(a\) is called the radical expression.
\(\bullet\) Based on the definition, expression \(\sqrt(-25)\), \(\sqrt(-4)\), etc. don't make sense.
Fact 2.
For quick calculations it will be useful to learn the table of squares natural numbers from \(1\) to \(20\) : \[\begin(array)(|ll|) \hline 1^2=1 & \quad11^2=121 \\ 2^2=4 & \quad12^2=144\\ 3^2=9 & \quad13 ^2=169\\ 4^2=16 & \quad14^2=196\\ 5^2=25 & \quad15^2=225\\ 6^2=36 & \quad16^2=256\\ 7^ 2=49 & \quad17^2=289\\ 8^2=64 & \quad18^2=324\\ 9^2=81 & \quad19^2=361\\ 10^2=100& \quad20^2= 400\\ \hline \end(array)\]
Fact 3.
What operations can you do with square roots?
\(\bullet\) The sum or difference of square roots IS NOT EQUAL to the square root of the sum or difference, that is \[\sqrt a\pm\sqrt b\ne \sqrt(a\pm b)\] Thus, if you need to calculate, for example, \(\sqrt(25)+\sqrt(49)\) , then initially you must find the values of \(\sqrt(25)\) and \(\sqrt(49)\ ) and then fold them. Hence, \[\sqrt(25)+\sqrt(49)=5+7=12\] If the values \(\sqrt a\) or \(\sqrt b\) cannot be found when adding \(\sqrt a+\sqrt b\), then such an expression is not transformed further and remains as it is. For example, in the sum \(\sqrt 2+ \sqrt (49)\) we can find \(\sqrt(49)\) is \(7\) , but \(\sqrt 2\) cannot be transformed in any way, That's why \(\sqrt 2+\sqrt(49)=\sqrt 2+7\). Unfortunately, this expression cannot be simplified further\(\bullet\) The product/quotient of square roots is equal to the square root of the product/quotient, that is \[\sqrt a\cdot \sqrt b=\sqrt(ab)\quad \text(s)\quad \sqrt a:\sqrt b=\sqrt(a:b)\] (provided that both sides of the equalities make sense)
Example: \(\sqrt(32)\cdot \sqrt 2=\sqrt(32\cdot 2)=\sqrt(64)=8\);
\(\sqrt(768):\sqrt3=\sqrt(768:3)=\sqrt(256)=16\);
\(\sqrt((-25)\cdot (-64))=\sqrt(25\cdot 64)=\sqrt(25)\cdot \sqrt(64)= 5\cdot 8=40\). \(\bullet\) Using these properties, it is convenient to find the square roots of large numbers by factoring them.
Let's look at an example. Let's find \(\sqrt(44100)\) . Since \(44100:100=441\) , then \(44100=100\cdot 441\) . According to the criterion of divisibility, the number \(441\) is divisible by \(9\) (since the sum of its digits is 9 and is divisible by 9), therefore, \(441:9=49\), that is, \(441=9\ cdot 49\) .
Thus we got: \[\sqrt(44100)=\sqrt(9\cdot 49\cdot 100)= \sqrt9\cdot \sqrt(49)\cdot \sqrt(100)=3\cdot 7\cdot 10=210\] Let's look at another example: \[\sqrt(\dfrac(32\cdot 294)(27))= \sqrt(\dfrac(16\cdot 2\cdot 3\cdot 49\cdot 2)(9\cdot 3))= \sqrt( \ dfrac(16\cdot4\cdot49)(9))=\dfrac(\sqrt(16)\cdot \sqrt4 \cdot \sqrt(49))(\sqrt9)=\dfrac(4\cdot 2\cdot 7)3 =\dfrac(56)3\]
\(\bullet\) Let's show how to enter numbers under the square root sign using the example of the expression \(5\sqrt2\) (short notation for the expression \(5\cdot \sqrt2\)). Since \(5=\sqrt(25)\) , then \
Note also that, for example,
1) \(\sqrt2+3\sqrt2=4\sqrt2\) ,
2) \(5\sqrt3-\sqrt3=4\sqrt3\)
3) \(\sqrt a+\sqrt a=2\sqrt a\) .
Why is this so? Let's explain using example 1). As you already understand, we cannot somehow transform the number \(\sqrt2\). Let's imagine that \(\sqrt2\) is some number \(a\) . Accordingly, the expression \(\sqrt2+3\sqrt2\) is nothing more than \(a+3a\) (one number \(a\) plus three more of the same numbers \(a\)). And we know that this is equal to four such numbers \(a\) , that is, \(4\sqrt2\) .
Fact 4.
\(\bullet\) They often say “you can’t extract the root” when you can’t get rid of the sign \(\sqrt () \ \) of the root (radical) when finding the value of a number. For example, you can take the root of the number \(16\) because \(16=4^2\) , therefore \(\sqrt(16)=4\) . But it is impossible to extract the root of the number \(3\), that is, to find \(\sqrt3\), because there is no number that squared will give \(3\) .
Such numbers (or expressions with such numbers) are irrational. For example, numbers \(\sqrt3, \ 1+\sqrt2, \ \sqrt(15)\) etc. are irrational.
Also irrational are the numbers \(\pi\) (the number “pi”, approximately equal to \(3.14\)), \(e\) (this number is called the Euler number, it is approximately equal to \(2.7\)) etc.
\(\bullet\) Please note that any number will be either rational or irrational. And together all rational and all irrational numbers form a set called a set of real numbers. This set is denoted by the letter \(\mathbb(R)\) .
This means that all the numbers that are on at the moment we know are called real numbers.
Fact 5.
\(\bullet\) The modulus of a real number \(a\) is a non-negative number \(|a|\) equal to the distance from the point \(a\) to \(0\) on the real line. For example, \(|3|\) and \(|-3|\) are equal to 3, since the distances from the points \(3\) and \(-3\) to \(0\) are the same and equal to \(3 \) .
\(\bullet\) If \(a\) is a non-negative number, then \(|a|=a\) .
Example: \(|5|=5\) ; \(\qquad |\sqrt2|=\sqrt2\) . \(\bullet\) If \(a\) is a negative number, then \(|a|=-a\) .
Example: \(|-5|=-(-5)=5\) ; \(\qquad |-\sqrt3|=-(-\sqrt3)=\sqrt3\).
They say that for negative numbers the modulus “eats” the minus, while positive numbers, as well as the number \(0\), are left unchanged by the modulus.
BUT This rule only applies to numbers. If under your modulus sign there is an unknown \(x\) (or some other unknown), for example, \(|x|\) , about which we do not know whether it is positive, zero or negative, then get rid of the modulus we can't. In this case, this expression remains the same: \(|x|\) . \(\bullet\) The following formulas hold: \[(\large(\sqrt(a^2)=|a|))\] \[(\large((\sqrt(a))^2=a)), \text( provided ) a\geqslant 0\] Very often the following mistake is made: they say that \(\sqrt(a^2)\) and \((\sqrt a)^2\) are one and the same. This is only true if \(a\) is a positive number or zero. But if \(a\) is a negative number, then this is false. It is enough to consider this example. Let's take instead of \(a\) the number \(-1\) . Then \(\sqrt((-1)^2)=\sqrt(1)=1\) , but the expression \((\sqrt (-1))^2\) does not exist at all (after all, it is impossible to use the root sign put negative numbers!).
Therefore, we draw your attention to the fact that \(\sqrt(a^2)\) is not equal to \((\sqrt a)^2\) ! Example: 1) \(\sqrt(\left(-\sqrt2\right)^2)=|-\sqrt2|=\sqrt2\), because \(-\sqrt2<0\)
;
\(\phantom(00000)\) 2) \((\sqrt(2))^2=2\) . \(\bullet\) Since \(\sqrt(a^2)=|a|\) , then \[\sqrt(a^(2n))=|a^n|\] (the expression \(2n\) denotes an even number)
That is, when taking the root of a number that is to some degree, this degree is halved.
Example:
1) \(\sqrt(4^6)=|4^3|=4^3=64\)
2) \(\sqrt((-25)^2)=|-25|=25\) (note that if the module is not supplied, it turns out that the root of the number is equal to \(-25\) ; but we remember , that by definition of a root this cannot happen: when extracting a root, we should always get a positive number or zero)
3) \(\sqrt(x^(16))=|x^8|=x^8\) (since any number to an even power is non-negative)
Fact 6.
How to compare two square roots?
\(\bullet\) For square roots it is true: if \(\sqrt a<\sqrt b\)
, то \(a
1) compare \(\sqrt(50)\) and \(6\sqrt2\) . First, let's transform the second expression into \(\sqrt(36)\cdot \sqrt2=\sqrt(36\cdot 2)=\sqrt(72)\). Thus, since \(50<72\)
, то и \(\sqrt{50}<\sqrt{72}\)
. Следовательно, \(\sqrt{50}<6\sqrt2\)
.
2) Between what integers is \(\sqrt(50)\) located?
Since \(\sqrt(49)=7\) , \(\sqrt(64)=8\) , and \(49<50<64\)
, то \(7<\sqrt{50}<8\)
, то есть число \(\sqrt{50}\)
находится между числами \(7\)
и \(8\)
.
3) Let's compare \(\sqrt 2-1\) and \(0.5\) . Let's assume that \(\sqrt2-1>0.5\) : \[\begin(aligned) &\sqrt 2-1>0.5 \ \big| +1\quad \text((add one to both sides))\\ &\sqrt2>0.5+1 \\big| \ ^2 \quad\text((squaring both sides))\\ &2>1.5^2\\ &2>2.25 \end(aligned)\] We see that we have obtained an incorrect inequality. Therefore, our assumption was incorrect and \(\sqrt 2-1<0,5\)
.
Note that adding a certain number to both sides of the inequality does not affect its sign. Multiplying/dividing both sides of an inequality by a positive number also does not affect its sign, but multiplying/dividing by a negative number reverses the sign of the inequality!
You can square both sides of an equation/inequality ONLY IF both sides are non-negative. For example, in the inequality from the previous example you can square both sides, in the inequality \(-3<\sqrt2\)
нельзя (убедитесь в этом сами)!
\(\bullet\) It should be remembered that \[\begin(aligned) &\sqrt 2\approx 1.4\\ &\sqrt 3\approx 1.7 \end(aligned)\] Knowing the approximate meaning of these numbers will help you when comparing numbers! \(\bullet\) In order to extract the root (if it can be extracted) from some large number that is not in the table of squares, you must first determine between which “hundreds” it is located, then – between which “tens”, and then determine the last digit of this number. Let's show how this works with an example.
Let's take \(\sqrt(28224)\) . We know that \(100^2=10\,000\), \(200^2=40\,000\), etc. Note that \(28224\) is between \(10\,000\) and \(40\,000\) . Therefore, \(\sqrt(28224)\) is between \(100\) and \(200\) .
Now let’s determine between which “tens” our number is located (that is, for example, between \(120\) and \(130\)). Also from the table of squares we know that \(11^2=121\) , \(12^2=144\) etc., then \(110^2=12100\) , \(120^2=14400 \) , \(130^2=16900\) , \(140^2=19600\) , \(150^2=22500\) , \(160^2=25600\) , \(170^2=28900 \) . So we see that \(28224\) is between \(160^2\) and \(170^2\) . Therefore, the number \(\sqrt(28224)\) is between \(160\) and \(170\) .
Let's try to determine the last digit. Let's remember what single-digit numbers, when squared, give \(4\) at the end? These are \(2^2\) and \(8^2\) . Therefore, \(\sqrt(28224)\) will end in either 2 or 8. Let's check this. Let's find \(162^2\) and \(168^2\) :
\(162^2=162\cdot 162=26224\)
\(168^2=168\cdot 168=28224\) .
Therefore, \(\sqrt(28224)=168\) . Voila!
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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.
Yandex.RTB R-A-339285-1
Rules for adding and subtracting square roots
Definition 1Operations 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 26 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 a large number of 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 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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