Simplifying online functions. Converting Expressions. Detailed Theory (2019)

Simplifying algebraic expressions is one of the keys to learning algebra and is an extremely useful skill for all mathematicians. Simplification allows you to reduce a complex or long expression to a simple expression that is easy to work with. Basic skills of simplification are good even for those who are not enthusiastic about mathematics. By following a few simple rules, you can simplify many of the most common types of algebraic expressions without any special mathematical knowledge.

Steps

Important Definitions

1. Similar members. These are members with a variable of the same order, members with the same variables, or free members (members that do not contain a variable). In other words, similar terms include the same variable to the same degree, include several of the same variables, or do not include a variable at all. The order of the terms in the expression does not matter.

   • For example, 3x 2 and 4x 2 are similar terms because they contain a second-order (to the second power) variable "x". However, x and x2 are not similar terms, since they contain the variable “x” of different orders (first and second). Likewise, -3yx and 5xz are not similar terms because they contain different variables.

2. Factorization. This is finding numbers whose product leads to the original number. Any original number can have several factors. For example, the number 12 can be factored into the following series of factors: 1 × 12, 2 × 6 and 3 × 4, so we can say that the numbers 1, 2, 3, 4, 6 and 12 are factors of the number 12. The factors are the same as the factors , that is, the numbers by which the original number is divided.

   • For example, if you want to factor the number 20, write it like this: 4×5.
   • Note that when factoring, the variable is taken into account. For example, 20x = 4(5x).
   • Prime numbers cannot be factored because they are only divisible by themselves and 1.

3. Remember and follow the order of operations to avoid mistakes.

   • Brackets
   • Degree
   • Multiplication
   • Division
   • Addition
   • Subtraction

   Bringing similar members

   1. Write down the expression. Simple algebraic expressions (those that don't contain fractions, roots, etc.) can be solved (simplified) in just a few steps.

      • For example, simplify the expression 1 + 2x - 3 + 4x.

   2. Define similar terms (terms with a variable of the same order, terms with the same variables, or free terms).

      • Find similar terms in this expression. The terms 2x and 4x contain a variable of the same order (first). Also, 1 and -3 are free terms (do not contain a variable). Thus, in this expression the terms 2x and 4x are similar, and the members 1 and -3 are also similar.

   3. Give similar terms. This means adding or subtracting them and simplifying the expression.

      • 2x + 4x = 6x
      • 1 - 3 = -2

   4. Rewrite the expression taking into account the given terms. You will get a simple expression with fewer terms. The new expression is equal to the original one.

      • In our example: 1 + 2x - 3 + 4x = 6x - 2, that is, the original expression is simplified and easier to work with.

   5. Follow the order of operations when bringing similar members. In our example, it was easy to provide similar terms. However, in the case of complex expressions in which terms are enclosed in parentheses and fractions and roots are present, it is not so easy to bring such terms. In these cases, follow the order of operations.

      • For example, consider the expression 5(3x - 1) + x((2x)/(2)) + 8 - 3x. Here it would be a mistake to immediately define 3x and 2x as similar terms and give them, because it is necessary to open the parentheses first. Therefore, perform the operations according to their order.
        • 5(3x-1) + x((2x)/(2)) + 8 - 3x
        • 15x - 5 + x(x) + 8 - 3x
        • 15x - 5 + x 2 + 8 - 3x. Now, when the expression contains only addition and subtraction operations, you can bring similar terms.
        • x 2 + (15x - 3x) + (8 - 5)
        • x 2 + 12x + 3

   Taking the multiplier out of brackets

   1. Find the greatest common divisor (GCD) of all the coefficients of the expression. GCD is the largest number by which all coefficients of the expression are divided.

      • For example, consider the equation 9x 2 + 27x - 3. In this case, GCD = 3, since any coefficient of this expression is divisible by 3.

   2. Divide each term of the expression by gcd. The resulting terms will contain smaller coefficients than in the original expression.

      • In our example, divide each term in the expression by 3.
        • 9x 2 /3 = 3x 2
        • 27x/3 = 9x
        • -3/3 = -1
        • The result was an expression 3x 2 + 9x - 1. It is not equal to the original expression.

   3. Write the original expression as equal to the product of gcd and the resulting expression. That is, enclose the resulting expression in brackets, and take the gcd out of the brackets.

      • In our example: 9x 2 + 27x - 3 = 3(3x 2 + 9x - 1)

   4. Simplifying fractional expressions by putting the factor out of brackets. Why simply put the multiplier out of brackets, as was done earlier? Then, to learn how to simplify complex expressions, such as fractional expressions. In this case, putting the factor out of brackets can help get rid of the fraction (from the denominator).

      • For example, consider the fractional expression (9x 2 + 27x - 3)/3. Use factoring out to simplify this expression.
        • Put the factor of 3 out of brackets (as you did earlier): (3(3x 2 + 9x - 1))/3
        • Notice that there is now a 3 in both the numerator and the denominator. This can be reduced to give the expression: (3x 2 + 9x – 1)/1
        • Since any fraction that has the number 1 in the denominator is simply equal to the numerator, the original fraction expression simplifies to: 3x 2 + 9x - 1.

   Additional simplification methods

4. Let's look at a simple example: √(90). The number 90 can be factored into the following factors: 9 and 10, and from 9 we can take the square root (3) and take 3 out from under the root.
   • √(90)
   • √(9×10)
   • √(9)×√(10)
   • 3×√(10)
   • 3√(10)

5. Simplifying expressions with powers. Some expressions contain operations of multiplication or division of terms with powers. In the case of multiplying terms with the same base, their powers are added; in the case of dividing terms with the same base, their powers are subtracted.

   • For example, consider the expression 6x 3 × 8x 4 + (x 17 /x 15). In the case of multiplication, add the powers, and in the case of division, subtract them.
     • 6x 3 × 8x 4 + (x 17 /x 15)
     • (6 × 8)x 3 + 4 + (x 17 - 15)
     • 48x 7 + x 2
   • The following is an explanation of the rules for multiplying and dividing terms with powers.
     • Multiplying terms with powers is equivalent to multiplying terms by themselves. For example, since x 3 = x × x × x and x 5 = x × x × x × x × x, then x 3 × x 5 = (x × x × x) × (x × x × x × x × x), or x 8 .
     • Likewise, dividing terms with degrees is equivalent to dividing terms by themselves. x 5 / x 3 = (x × x × x × x × x)/(x × x × x). Since similar terms found in both the numerator and the denominator can be reduced, the product of two “x”, or x 2 , remains in the numerator.

• Always remember about the signs (plus or minus) preceding the terms of the expression, as many people have difficulty choosing the correct sign.
• Ask for help if needed!
• Simplifying algebraic expressions isn't easy, but once you get the hang of it, it's a skill you can use for the rest of your life.

Using any language, you can express the same information in different words and phrases. Mathematical language is no exception. But the same expression can be equivalently written in different ways. And in some situations, one of the entries is simpler. We'll talk about simplifying expressions in this lesson.

People communicate in different languages. For us, an important comparison is the pair “Russian language - mathematical language”. The same information can be communicated in different languages. But, besides this, it can be pronounced in different ways in one language.

For example: “Petya is friends with Vasya”, “Vasya is friends with Petya”, “Petya and Vasya are friends”. Said differently, but the same thing. From any of these phrases we would understand what we are talking about.

Let's look at this phrase: “The boy Petya and the boy Vasya are friends.” We understand what we are talking about. However, we don't like the sound of this phrase. Can't we simplify it, say the same thing, but simpler? “Boy and boy” - you can say once: “The boys Petya and Vasya are friends.”

“Boys”... Isn’t it clear from their names that they are not girls? We remove the “boys”: “Petya and Vasya are friends.” And the word “friends” can be replaced with “friends”: “Petya and Vasya are friends.” As a result, the first, long, ugly phrase was replaced with an equivalent statement that is easier to say and easier to understand. We have simplified this phrase. To simplify means to say it more simply, but not to lose or distort the meaning.

In mathematical language, approximately the same thing happens. The same thing can be said, written differently. What does it mean to simplify an expression? This means that for the original expression there are many equivalent expressions, that is, those that mean the same thing. And from all this variety we must choose the simplest, in our opinion, or the most suitable for our further purposes.

For example, consider the numeric expression . It will be equivalent to .

It will also be equivalent to the first two: .

It turns out that we have simplified our expressions and found the shortest equivalent expression.

For numeric expressions, you always need to do everything and get the equivalent expression as a single number.

Let's look at an example of a literal expression . Obviously, it will be simpler.

When simplifying literal expressions, it is necessary to perform all possible actions.

Is it always necessary to simplify an expression? No, sometimes it will be more convenient for us to have an equivalent but longer entry.

Example: you need to subtract a number from a number.

It is possible to calculate, but if the first number were represented by its equivalent notation: , then the calculations would be instantaneous: .

That is, a simplified expression is not always beneficial for us for further calculations.

Nevertheless, very often we are faced with a task that just sounds like “simplify the expression.”

Simplify the expression: .

Solution

1) Perform the actions in the first and second brackets: .

2) Let's calculate the products: .

Obviously, the last expression has a simpler form than the initial one. We've simplified it.

In order to simplify the expression, it must be replaced with an equivalent (equal).

To determine the equivalent expression you need:

1) perform all possible actions,

2) use the properties of addition, subtraction, multiplication and division to simplify calculations.

Properties of addition and subtraction:

1. Commutative property of addition: rearranging the terms does not change the sum.

2. Combinative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third numbers to the first number.

3. The property of subtracting a sum from a number: to subtract a sum from a number, you can subtract each term separately.

Properties of multiplication and division

1. Commutative property of multiplication: rearranging the factors does not change the product.

2. Combinative property: to multiply a number by the product of two numbers, you can first multiply it by the first factor, and then multiply the resulting product by the second factor.

3. Distributive property of multiplication: in order to multiply a number by a sum, you need to multiply it by each term separately.

Let's see how we actually do mental calculations.

Calculate:

Solution

1) Let's imagine how

2) Let's imagine the first factor as a sum of bit terms and perform the multiplication:

3) you can imagine how and perform multiplication:

4) Replace the first factor with an equivalent sum:

The distribution law can also be used in the opposite direction: .

Follow these steps:

1) 2)

Solution

1) For convenience, you can use the distributive law, but use it in the opposite direction - take the common factor out of brackets.

2) Let’s take the common factor out of brackets

It is necessary to buy linoleum for the kitchen and hallway. Kitchen area - , hallway - . There are three types of linoleums: for, and rubles for. How much will each of the three types of linoleum cost? (Fig. 1)

Rice. 1. Illustration for the problem statement

Solution

Method 1. You can separately find out how much money it will take to buy linoleum for the kitchen, and then put it in the hallway and add up the resulting products.

Note 1

A Boolean function can be written using a Boolean expression and can then be moved to a logic circuit. It is necessary to simplify logical expressions in order to obtain the simplest possible (and therefore cheaper) logical circuit. In fact, a logical function, a logical expression and a logical circuit are three different languages ​​that talk about one entity.

To simplify logical expressions use laws of algebra logic.

Some transformations are similar to transformations of formulas in classical algebra (taking the common factor out of brackets, using commutative and combinational laws, etc.), while other transformations are based on properties that the operations of classical algebra do not have (using the distributive law for conjunction, laws of absorption, gluing, de Morgan's rules, etc.).

The laws of logical algebra are formulated for basic logical operations - “NOT” – inversion (negation), “AND” – conjunction (logical multiplication) and “OR” – disjunction (logical addition).

The law of double negation means that the “NOT” operation is reversible: if you apply it twice, then in the end the logical value will not change.

The law of excluded middle states that any logical expression is either true or false (“there is no third”). Therefore, if $A=1$, then $\bar(A)=0$ (and vice versa), which means that the conjunction of these quantities is always equal to zero, and the disjunction is always equal to one.

$((A + B) → C) \cdot (B → C \cdot D) \cdot C.$

Let's simplify this formula:

Figure 3.

It follows that $A = 0$, $B = 1$, $C = 1$, $D = 1$.

Answer: Students $B$, $C$ and $D$ play chess, but student $A$ does not play.

When simplifying logical expressions, you can perform the following sequence of actions:

1. Replace all “non-basic” operations (equivalence, implication, exclusive OR, etc.) with their expressions through the basic operations of inversion, conjunction and disjunction.
2. Expand inversions of complex expressions according to De Morgan's rules in such a way that negation operations remain only for individual variables.
3. Then simplify the expression using opening brackets, placing common factors outside brackets and other laws of logical algebra.

Example 2

Here, De Morgan's rule, the distributive law, the law of the excluded middle, the commutative law, the law of repetition, again the commutative law and the law of absorption are used successively.

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:

1. brackets;
2. exponentiation;
3. multiplication;
4. division;
5. addition;
6. 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! Even if the answer is correct, it will be 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. Thus, 20 can be represented as 20x1, 2x10, 5x4, 2x5x2, or in 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 are carried outside 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:

1. If there is a multiplication sign between the variables, the powers add up.
2. 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.

Video

This video will help you understand and remember how expressions are simplified.

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Mathematical-Calculator-Online v.1.0

The calculator performs the following operations: addition, subtraction, multiplication, division, working with decimals, root extraction, exponentiation, percent calculation and other operations.

Solution:

How to use a math calculator

Algorithm of the online calculator using examples

Addition.

Addition of natural integers (5 + 7 = 12)

Addition of integer natural and negative numbers ( 5 + (-2) = 3 )

Adding decimal fractions (0.3 + 5.2 = 5.5)

Subtraction.

Subtracting natural integers ( 7 - 5 = 2 )

Subtracting natural and negative integers ( 5 - (-2) = 7 )

Subtracting decimal fractions (6.5 - 1.2 = 4.3)

Multiplication.

Product of natural integers (3 * 7 = 21)

Product of natural and negative integers ( 5 * (-3) = -15 )

Product of decimal fractions ( 0.5 * 0.6 = 0.3 )

Division.

Division of natural integers (27 / 3 = 9)

Division of natural and negative integers (15 / (-3) = -5)

Division of decimal fractions (6.2 / 2 = 3.1)

Extracting the root of a number.

Extracting the root of an integer ( root(9) = 3)

Extracting the root of decimal fractions (root(2.5) = 1.58)

Extracting the root of a sum of numbers ( root(56 + 25) = 9)

Extracting the root of the difference between numbers (root (32 – 7) = 5)

Squaring a number.

Squaring an integer ( (3) 2 = 9 )

Squaring decimals ((2,2)2 = 4.84)

Conversion to decimal fractions.

Calculating percentages of a number

Increase the number 230 by 15% ( 230 + 230 * 0.15 = 264.5 )

Reduce the number 510 by 35% ( 510 – 510 * 0.35 = 331.5 )

18% of the number 140 is (140 * 0.18 = 25.2)