Simplifying algebraic fractions online calculator. How to simplify algebraic expressions
An algebraic expression in which, along with the operations of addition, subtraction and multiplication, they also use division by literal expressions, is called a fractional algebraic expression. Such are, for example, the expressions
We call an algebraic fraction an algebraic expression that has the form of a quotient of division of two integer algebraic expressions (for example, monomials or polynomials). Such are, for example, the expressions
the third of the expressions).
Identity transformations of fractional algebraic expressions are for the most part intended to represent them as an algebraic fraction. To find a common denominator, the factorization of the denominators of fractions - terms is used in order to find their least common multiple. When reducing algebraic fractions the strict identity of expressions may be violated: it is necessary to exclude the values of quantities at which the factor by which the reduction is made vanishes.
Here are some examples identical transformations fractional algebraic expressions.
Example 1: Simplify an expression
All terms can be reduced to a common denominator (it is convenient to change the sign in the denominator of the last term and the sign in front of it):
Our expression is equal to one for all values except these values, it is not defined and fraction reduction is illegal).
Example 2. Represent expression as an algebraic fraction
Solution. The expression can be taken as a common denominator. We find successively:
Exercises
1. Find the values of algebraic expressions for the specified values of the parameters:
2. Factorize.
Math-Calculator-Online v.1.0
The calculator performs the following operations: addition, subtraction, multiplication, division, working with decimals, extracting the root, raising to a power, calculating percentages, and other operations.
Solution:
How to use the math calculator
Key | Designation | Explanation |
---|---|---|
5 | numbers 0-9 | Arabic numerals. Enter natural integers, zero. To get a negative integer, press the +/- key |
. | semicolon) | A decimal separator. If there is no digit before the dot (comma), the calculator will automatically substitute a zero before the dot. For example: .5 - 0.5 will be written |
+ | plus sign | Addition of numbers (whole, decimal fractions) |
- | minus sign | Subtraction of numbers (whole, decimal fractions) |
÷ | division sign | Division of numbers (whole, decimal fractions) |
X | multiplication sign | Multiplication of numbers (integers, decimals) |
√ | root | Extracting the root from a number. When you press the "root" button again, the root is calculated from the result. For example: square root of 16 = 4; square root of 4 = 2 |
x2 | squaring | Squaring a number. When you press the "squaring" button again, the result is squared. For example: square 2 = 4; square 4 = 16 |
1/x | fraction | Output to decimals. In the numerator 1, in the denominator the input number |
% | percent | Get a percentage of a number. To work, you must enter: the number from which the percentage will be calculated, the sign (plus, minus, divide, multiply), how many percent in numerical form, the "%" button |
( | open bracket | An open parenthesis to set the evaluation priority. A closed parenthesis is required. Example: (2+3)*2=10 |
) | closed bracket | A closed parenthesis to set the evaluation priority. Availability required open bracket |
± | plus minus | Changes sign to opposite |
= | equals | Displays the result of the solution. Also, intermediate calculations and the result are displayed above the calculator in the "Solution" field. |
← | deleting a character | Deletes the last character |
FROM | reset | Reset button. Completely resets the calculator to "0" |
The algorithm of the online calculator with examples
Addition.
Integer addition natural numbers { 5 + 7 = 12 }
Addition of whole natural and negative numbers ( 5 + (-2) = 3 )
Decimal addition fractional numbers { 0,3 + 5,2 = 5,5 }
Subtraction.
Subtraction of whole natural numbers ( 7 - 5 = 2 )
Subtraction of whole natural and negative numbers ( 5 - (-2) = 7 )
Subtraction of decimal fractional numbers ( 6.5 - 1.2 = 4.3 )
Multiplication.
Product of whole natural numbers ( 3 * 7 = 21 )
Product of whole natural and negative numbers ( 5 * (-3) = -15 )
Product of decimal fractional numbers ( 0.5 * 0.6 = 0.3 )
Division.
Division of whole natural numbers ( 27 / 3 = 9 )
Division of whole natural and negative numbers ( 15 / (-3) = -5 )
Division of decimal fractional numbers ( 6.2 / 2 = 3.1 )
Extracting the root from a number.
Extracting the root of an integer ( root(9) = 3 )
Extracting the root of decimals ( root(2.5) = 1.58 )
Extracting the root from the sum of numbers ( root(56 + 25) = 9 )
Extracting the root of the difference in numbers ( root (32 - 7) = 5 )
Squaring a number.
Squaring an integer ( (3) 2 = 9 )
Squaring decimals ( (2.2) 2 = 4.84 )
Convert to decimal fractions.
Calculating percentages of a number
Increase 230 by 15% ( 230 + 230 * 0.15 = 264.5 )
Decrease the number 510 by 35% ( 510 - 510 * 0.35 = 331.5 )
18% of the number 140 is ( 140 * 0.18 = 25.2 )
Some algebraic examples one kind is capable of terrifying schoolchildren. Long expressions are not only intimidating, but also very difficult to calculate. Trying to immediately understand what follows and what follows, not to get confused for long. It is for this reason that mathematicians always try to simplify the “terrible” task as much as possible and only then proceed to solve it. Oddly enough, such a trick greatly speeds up the process.
Simplification is one of the fundamental points in algebra. If in simple tasks you can still do without it, 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 how to put into 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 operations with numbers:
- parentheses;
- exponentiation;
- multiplication;
- division;
- addition;
- subtraction.
The last two points can be safely swapped and this will not affect the result in any way. But adding two neighboring numbers, when next to one of them there is a multiplication sign, is absolutely impossible! The answer, if any, is wrong. 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 members that do not have next to them the letter designation of the unknown.
The bottom line is that in the absence of parentheses You can simplify the expression by adding or subtracting like.
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 are simplified to (8+3)x 2 =11x 2, while when subtracted, it turns out (8-3)x 2 =5x 2;
- 4x 3 and 6x - and here "x" has a different degree;
- 2y 7 and 33x 7 - contain different variables, therefore, as in the previous case, they do not belong to similar ones.
Factoring a Number
This little mathematical trick, if you learn how to use it correctly, will help you to cope with a tricky problem more than once in the future. And it’s easy to understand how the “system” works: a decomposition is a product of several elements, the calculation of which gives the original value. Thus, 20 can be represented as 20x1, 2x10, 5x4, 2x5x2, or some other way.
On a note: multipliers are always the same as divisors. So you need to look for a working “pair” for expansion among the numbers by which the original is divisible without a remainder.
You can perform such an operation both with free members and with digits attached to a variable. The main thing is not to lose the latter during calculations - even after decomposition, the unknown cannot take and "go nowhere." It remains at one of the factors:
- 15x=3(5x);
- 60y 2 \u003d (15y 2) 4.
Prime numbers that can only be divided by themselves or 1 never factor - it makes no sense..
Basic Simplification Methods
The first thing that catches the eye:
- the presence of brackets;
- fractions;
- roots.
Algebraic examples in school curriculum are often compiled with the assumption that they can be beautifully simplified.
Bracket Calculations
Pay close attention to the sign in front of the brackets! Multiplication or division is applied to each element inside, and minus - reverses the existing "+" or "-" signs.
Parentheses are calculated according to the rules or according to the formulas of abbreviated multiplication, after which similar ones are given.
Fraction reduction
Reduce fractions is also easy. They themselves “willingly run away” once in a while, it is worth making operations with bringing such members. But you can simplify the example even before this: 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 delete the superfluous, in the second you will have to think, bringing part of the expression to the form for simplification. Methods used:
- search and bracketing of 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 simplification problem is almost the same as 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 interfering with calculations. For example, to unobtrusive √(3) or √(7).
The right way simplify the radical expression - try to factor it, some of which are 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 a numerator or denominator;
- it is impossible to decompose and take out a part of the sum or difference 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 for the possibility of rendering: √(x 2 y)=x√(y), √(x 3)=√(x 2 ×x)=x√( x);
- sometimes it is allowed to get rid of the radical variable by raising it to a fractional power: √ (y 3)=y 3/2.
Power Expression Simplification
If in the case of simple calculations by minus or plus, examples are simplified by bringing similar ones, then what about when multiplying or dividing variables with varying degrees? They can be easily simplified by remembering two main points:
- If there is a multiplication sign between the variables, the exponents are added.
- When they are divided by each other, the same denominator is subtracted from the degree of the numerator.
The only condition for such a simplification is same base for both members. Examples for clarity:
- 5x 2 × 4x 7 + (y 13 / y 11) \u003d (5 × 4)x 2+7 + y 13- 11 \u003d 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 numerical 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 member to a power means multiplying it by itself a certain number of times, i.e. x 2 \u003d x × x;
- division is similar: if you expand the degree of the numerator and denominator, then some of the variables will be reduced, while the rest are “gathered”, which is equivalent to subtraction.
As in any business, when simplifying algebraic expressions, not only knowledge of the basics is necessary, but also practice. After just a few lessons, examples that once seemed complicated will be reduced without special work, turning into short and easily solved.
Video
This video will help you understand and remember how expressions are simplified.
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