Solve the expression online calculator with solution. Power expressions (expressions with powers) and their transformation
Some algebraic examples alone can terrify schoolchildren. Long expressions are not only intimidating, but also make calculations very difficult. Trying to immediately understand what follows what, it won’t take long to get confused. It is for this reason that mathematicians always try to simplify a “terrible” problem as much as possible and only then begin to solve it. Oddly enough, this trick significantly speeds up the work process.
Simplification is one of the fundamental points in algebra. If you can still do without it in simple problems, then more difficult to calculate examples may turn out to be too tough. This is where these skills come in handy! Moreover, complex mathematical knowledge is not required: it will be enough just to remember and learn to apply in practice a few basic techniques and formulas.
Regardless of the complexity of the calculations, when solving any expression it is important follow the order of performing operations with numbers:
- brackets;
- exponentiation;
- multiplication;
- division;
- addition;
- subtraction.
The last two points can be easily swapped and this will not affect the result in any way. But adding two adjacent numbers when there is a multiplication sign next to one of them is absolutely forbidden! The answer, if any, is incorrect. Therefore, you need to remember the sequence.
The use of such
Such elements include numbers with a variable of the same order or the same degree. There are also so-called free terms that do not have a letter designation for the unknown next to them.
The point is that in the absence of parentheses you can simplify the expression by adding or subtracting similar.
A few illustrative examples:
- 8x 2 and 3x 2 - both numbers have the same second-order variable, so they are similar and when added they simplify to (8+3)x 2 =11x 2, while when subtracted they get (8-3)x 2 =5x 2 ;
- 4x 3 and 6x - and here “x” has different degrees;
- 2y 7 and 33x 7 - contain different variables, therefore, as in the previous case, they are not similar.
Factoring a number
This little mathematical trick, if you learn to use it correctly, will more than once help you cope with a tricky problem in the future. And it’s not difficult to understand how the “system” works: decomposition is the product of several elements, the calculation of which gives the original value. So 20 can be represented as 20x1, 2x10, 5x4, 2x5x2, or some other way.
On a note: Factors are always the same as divisors. So you need to look for a working “pair” for decomposition among the numbers into which the original is divisible without a remainder.
This operation can be performed both with free terms and with numbers in a variable. The main thing is not to lose the latter during calculations - even after decomposition, the unknown cannot just “go nowhere.” It remains at one of the multipliers:
- 15x=3(5x);
- 60y 2 = (15y 2)4.
Prime numbers that can only be divided by themselves or 1 are never expanded - it makes no sense.
Basic methods of simplification
The first thing your eye catches:
- the presence of parentheses;
- fractions;
- roots.
Algebraic examples in the school curriculum are often written with the idea that they can be beautifully simplified.
Calculations in parentheses
Pay close attention to the sign in front of the brackets! Multiplication or division is applied to each element inside, and a minus sign reverses the existing “+” or “-” signs.
Brackets are calculated according to the rules or using abbreviated multiplication formulas, after which similar ones are given.
Reducing Fractions
Reduce fractions It's also easy. They themselves “willingly run away” every once in a while, as soon as operations are carried out to bring in such members. But you can simplify the example even before that: pay attention to the numerator and denominator. They often contain explicit or hidden elements that can be mutually reduced. True, if in the first case you just need to cross out the unnecessary, in the second you will have to think, bringing part of the expression to form for simplification. Methods used:
- searching for and bracketing the greatest common divisor of the numerator and denominator;
- dividing each top element by the denominator.
When an expression or part of it is under the root, the primary task of simplification is almost similar to the case with fractions. It is necessary to look for ways to completely get rid of it or, if this is not possible, to minimize the sign that interferes with calculations. For example, up to the unobtrusive √(3) or √(7).
A surefire way to simplify a radical expression is to try to factor it, some of which extend beyond the sign. An illustrative example: √(90)=√(9×10) =√(9)×√(10)=3√(10).
Other little tricks and nuances:
- this simplification operation can be carried out with fractions, taking it out of the sign both as a whole and separately as the numerator or denominator;
- Part of the sum or difference cannot be expanded and taken beyond the root;
- when working with variables, be sure to take into account its degree, it must be equal to or a multiple of the root to be able to be taken out: √(x 2 y)=x√(y), √(x 3)=√(x 2 ×x)=x√( x);
- sometimes it is possible to get rid of the radical variable by raising it to a fractional power: √(y 3)=y 3/2.
Simplifying a Power Expression
If in the case of simple calculations by minus or plus, examples are simplified by citing similar ones, then what about when multiplying or dividing variables with different powers? They can be easily simplified by remembering two main points:
- If there is a multiplication sign between the variables, the powers add up.
- When they are divided by each other, the same power of the denominator is subtracted from the power of the numerator.
The only condition for such simplification is that both terms have the same basis. Examples for clarity:
- 5x 2 ×4x 7 +(y 13 /y 11)=(5×4)x 2+7 +y 13- 11 =20x 9 +y 2;
- 2z 3 +z×z 2 -(3×z 8 /z 5)=2z 3 +z 1+2 -(3×z 8-5)=2z 3 +z 3 -3z 3 =3z 3 -3z 3 = 0.
We note that operations with numeric values in front of variables occur according to the usual mathematical rules. And if you look closely, it becomes clear that the power elements of the expression “work” in a similar way:
- raising a term to a power means multiplying it by itself a certain number of times, i.e. x 2 =x×x;
- division is similar: if you expand the powers of the numerator and denominator, then some of the variables will be canceled, while the remaining ones are “collected,” which is equivalent to subtraction.
As with anything, simplifying algebraic expressions requires not only knowledge of the basics, but also practice. After just a few lessons, examples that once seemed complex will be reduced without much difficulty, turning into short and easily solved ones.
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Convenient and simple online fraction calculator with detailed solutions Maybe:
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The result of solving fractions will be here...
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Fraction sign "/" + - * :
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Our online fraction calculator has quick input. To solve fractions, for example, simply write 1/2+2/7
into the calculator and press the " Solve fractions". The calculator will write to you detailed solution of fractions and will issue an easy-to-copy image.
Signs used for writing in a calculator
You can type an example for a solution either from the keyboard or using buttons.
Features of the online fraction calculator
The fraction calculator can only perform operations on 2 simple fractions. They can be either correct (the numerator is less than the denominator) or incorrect (the numerator is greater than the denominator). The numbers in the numerator and denominators cannot be negative or greater than 999.Our online calculator solves fractions and brings the answer to the correct form - it reduces the fraction and selects the whole part, if necessary.
If you need to solve negative fractions, just use the properties of minus. When multiplying and dividing negative fractions, minus by minus gives plus. That is, the product and division of negative fractions is equal to the product and division of the same positive ones. If one fraction is negative when multiplying or dividing, then simply remove the minus and then add it to the answer. When adding negative fractions, the result will be the same as if you were adding the same positive fractions. If you add one negative fraction, then this is the same as subtracting the same positive one.
When subtracting negative fractions, the result will be the same as if they were swapped and made positive. That is, minus by minus in this case gives a plus, but rearranging the terms does not change the sum. We use the same rules when subtracting fractions, one of which is negative.
To solve mixed fractions (fractions in which the whole part is isolated), simply fit the whole part into the fraction. To do this, multiply the whole part by the denominator and add to the numerator.
If you need to solve 3 or more fractions online, you should solve them one by one. First, count the first 2 fractions, then solve the next fraction with the answer you get, and so on. Perform the operations one by one, 2 fractions at a time, and eventually you will get the correct answer.
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We often hear this unpleasant phrase: “simplify the expression.” Usually we see some kind of monster like this:
“It’s much simpler,” we say, but such an answer usually doesn’t work.
Now I will teach you not to be afraid of any such tasks.
Moreover, at the end of the lesson, you yourself will simplify this example to (just!) an ordinary number (yes, to hell with these letters).
But before you start this activity, you need to be able to handle fractions And factor polynomials.
Therefore, if you have not done this before, be sure to master the topics “” and “”.
Have you read it? If yes, then you are now ready.
Let's go! (Let's go!)
Basic Expression Simplification Operations
Now let's look at the basic techniques that are used to simplify expressions.
The simplest one is
1. Bringing similar
What are similar? You took this in 7th grade, when letters instead of numbers first appeared in mathematics.
Similar- these are terms (monomials) with the same letter part.
For example, in the sum, similar terms are and.
Do you remember?
Give similar- means adding several similar terms to each other and getting one term.
How can we put the letters together? - you ask.
This is very easy to understand if you imagine that the letters are some kind of objects.
For example, a letter is a chair. Then what is the expression equal to?
Two chairs plus three chairs, how many will it be? That's right, chairs: .
Now try this expression: .
To avoid confusion, let different letters represent different objects.
For example, - is (as usual) a chair, and - is a table.
chairs tables chair tables chairs chairs tables
The numbers by which the letters in such terms are multiplied are called coefficients.
For example, in a monomial the coefficient is equal. And in it is equal.
So, the rule for bringing similar ones is:
Examples:
Give similar ones:
Answers:
2. (and similar, since, therefore, these terms have the same letter part).
2. Factorization
This is usually the most important part in simplifying expressions.
After you have given similar ones, most often the resulting expression is needed factorize, that is, presented in the form of a product.
Especially this important in fractions: after all, in order to be able to reduce the fraction, The numerator and denominator must be represented as a product.
You went through the methods of factoring expressions in detail in the topic “”, so here you just have to remember what you learned.
To do this, solve several examples (you need to factorize them)
Examples:
Solutions:
3. Reducing a fraction.
Well, what could be more pleasant than crossing out part of the numerator and denominator and throwing them out of your life?
That's the beauty of downsizing.
It's simple:
If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.
This rule follows from the basic property of a fraction:
That is, the essence of the reduction operation is that We divide the numerator and denominator of the fraction by the same number (or by the same expression).
To reduce a fraction you need:
1) numerator and denominator factorize
2) if the numerator and denominator contain common factors, they can be crossed out.
Examples:
The principle, I think, is clear?
I would like to draw your attention to one typical mistake when abbreviating. Although this topic is simple, many people do everything wrong, not understanding that reduce- this means divide numerator and denominator are the same number.
No abbreviations if the numerator or denominator is a sum.
For example: we need to simplify.
Some people do this: which is absolutely wrong.
Another example: reduce.
The “smartest” will do this:
Tell me what's wrong here? It would seem: - this is a multiplier, which means it can be reduced.
But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not factorized.
Here's another example: .
This expression is factorized, which means you can reduce it, that is, divide the numerator and denominator by, and then by:
You can immediately divide it into:
To avoid such mistakes, remember an easy way to determine whether an expression is factorized:
The arithmetic operation that is performed last when calculating the value of an expression is the “master” operation.
That is, if you substitute some (any) numbers instead of letters and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is factorized).
If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be reduced).
To reinforce this, solve a few examples yourself:
Examples:
Solutions:
4. Adding and subtracting fractions. Reducing fractions to a common denominator.
Adding and subtracting ordinary fractions is a familiar operation: we look for a common denominator, multiply each fraction by the missing factor and add/subtract the numerators.
Let's remember:
Answers:
1. The denominators and are relatively prime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:
2. Here the common denominator is:
3. Here, first of all, we convert mixed fractions into improper ones, and then according to the usual scheme:
It's a completely different matter if the fractions contain letters, for example:
Let's start with something simple:
a) Denominators do not contain letters
Here everything is the same as with ordinary numerical fractions: we find the common denominator, multiply each fraction by the missing factor and add/subtract the numerators:
Now in the numerator you can give similar ones, if any, and factor them:
Try it yourself:
Answers:
b) Denominators contain letters
Let's remember the principle of finding a common denominator without letters:
· first of all, we determine the common factors;
· then we write out all the common factors one at a time;
· and multiply them by all other non-common factors.
To determine the common factors of the denominators, we first factor them into prime factors:
Let us emphasize the common factors:
Now let’s write out the common factors one at a time and add to them all the non-common (not underlined) factors:
This is the common denominator.
Let's get back to the letters. The denominators are given in exactly the same way:
· factor the denominators;
· determine common (identical) factors;
· write out all common factors once;
· multiply them by all other non-common factors.
So, in order:
1) factor the denominators:
2) determine common (identical) factors:
3) write out all common factors once and multiply them by all other (unemphasized) factors:
So there's a common denominator here. The first fraction must be multiplied by, the second - by:
By the way, there is one trick:
For example: .
We see the same factors in the denominators, only all with different indicators. The common denominator will be:
to a degree
to a degree
to a degree
to a degree.
Let's complicate the task:
How to make fractions have the same denominator?
Let's remember the basic property of a fraction:
Nowhere does it say that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!
See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What did you learn?
So, another unshakable rule:
When you reduce fractions to a common denominator, use only the multiplication operation!
But what do you need to multiply by to get?
So multiply by. And multiply by:
We will call expressions that cannot be factorized “elementary factors.”
For example, - this is an elementary factor. - Same. But no: it can be factorized.
What about the expression? Is it elementary?
No, because it can be factorized:
(you already read about factorization in the topic “”).
So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will deal with them in the same way.
We see that both denominators have a multiplier. It will go to the common denominator to the degree (remember why?).
The factor is elementary, and they do not have a common factor, which means that the first fraction will simply have to be multiplied by it:
Another example:
Solution:
Before you multiply these denominators in a panic, you need to think about how to factor them? They both represent:
Great! Then:
Another example:
Solution:
As usual, let's factorize the denominators. In the first denominator we simply put it out of brackets; in the second - the difference of squares:
It would seem that there are no common factors. But if you look closely, they are similar... And it’s true:
So let's write:
That is, it turned out like this: inside the bracket we swapped the terms, and at the same time the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.
Now let's bring it to a common denominator:
Got it? Let's check it now.
Tasks for independent solution:
Answers:
5. Multiplication and division of fractions.
Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:
Procedure
What is the procedure for calculating a numerical expression? Remember by calculating the meaning of this expression:
Did you count?
It should work.
So, let me remind you.
The first step is to calculate the degree.
The second is multiplication and division. If there are several multiplications and divisions at the same time, they can be done in any order.
And finally, we perform addition and subtraction. Again, in any order.
But: the expression in brackets is evaluated out of turn!
If several brackets are multiplied or divided by each other, we first calculate the expression in each of the brackets, and then multiply or divide them.
What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. When calculating an expression, what should you do first? That's right, calculate the brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.
So, the procedure for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):
Okay, it's all simple.
But this is not the same as an expression with letters?
No, it's the same! Only instead of arithmetic operations, you need to do algebraic ones, that is, the actions described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use this when working with fractions). Most often, to factorize, you need to use I or simply put the common factor out of brackets.
Usually our goal is to represent the expression as a product or quotient.
For example:
Let's simplify the expression.
1) First, we simplify the expression in brackets. There we have a difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:
It is impossible to simplify this expression any further; all the factors here are elementary (do you still remember what this means?).
2) We get:
Multiplying fractions: what could be simpler.
3) Now you can shorten:
OK it's all over Now. Nothing complicated, right?
Another example:
Simplify the expression.
First, try to solve it yourself, and only then look at the solution.
Solution:
First of all, let's determine the order of actions.
First, let's add the fractions in parentheses, so instead of two fractions we get one.
Then we will do division of fractions. Well, let's add the result with the last fraction.
I will number the steps schematically:
Finally, I will give you two useful tips:
1. If there are similar ones, they must be brought immediately. At whatever point similar ones arise in our country, it is advisable to bring them up immediately.
2. The same applies to reducing fractions: as soon as the opportunity to reduce appears, it must be taken advantage of. The exception is for fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.
Here are some tasks for you to solve on your own:
And what was promised at the very beginning:
Answers:
Solutions (brief):
If you have coped with at least the first three examples, then you have mastered the topic.
Now on to learning!
CONVERTING EXPRESSIONS. SUMMARY AND BASIC FORMULAS
Basic simplification operations:
- Bringing similar: to add (reduce) similar terms, you need to add their coefficients and assign the letter part.
- Factorization: putting the common factor out of brackets, applying it, etc.
- Reducing a fraction: The numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
1) numerator and denominator factorize
2) if the numerator and denominator have common factors, they can be crossed out.IMPORTANT: only multipliers can be reduced!
- Adding and subtracting fractions:
; - Multiplying and dividing fractions:
;
Well, the topic is over. If you are reading these lines, it means you are very cool.
Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!
Now the most important thing.
You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough...
For what?
For successfully passing the Unified State Exam, for entering college on a budget and, MOST IMPORTANTLY, for life.
I won’t convince you of anything, I’ll just say one thing...
People who have received a good education earn much more than those who have not received it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...
But think for yourself...
What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?
GAIN YOUR HAND BY SOLVING PROBLEMS ON THIS TOPIC.
You won't be asked for theory during the exam.
You will need solve problems against time.
And, if you haven’t solved them (A LOT!), you’ll definitely make a stupid mistake somewhere or simply won’t have time.
It's like in sports - you need to repeat it many times to win for sure.
Find the collection wherever you want, necessarily with solutions, detailed analysis and decide, decide, decide!
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If you don't like our tasks, find others. Just don't stop at theory.
“Understood” and “I can solve” are completely different skills. You need both.
Find problems and solve them!
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