First, the definition of score:
Generally, if A and B represent two integers and B contains letters, the formula
B is the denominator.
Second, the conditions related to the score
① The score is meaningful: the denominator is not 0(B0)
② The score is meaningless: the denominator is 0(B0)
③ Fraction value is 0: numerator is 0, denominator is not 0(A is called fraction, A is numerator, BA0).
B0
A0A0 or) B0B0 B0
A0A0 or)
B0B0④ Fractional value is positive or greater than 0: numerator and denominator have the same sign (⑤ Fractional value is negative or less than 0: numerator and denominator have different signs (
⑥ The fractional value is 1: the numerator and denominator are equal (A=B).
⑦ The fractional value is-1: the numerator and denominator values are reciprocal (A+B=0).
Third, the basic nature of the score
The numerator and denominator of a fraction are multiplied (or divided) by an algebraic expression that is not equal to 0, and the value of the fraction remains unchanged. The letter stands for AACAAC, where A, B and C are algebraic expressions and C0, respectively. BBCBBC
Extension: Symbolic Law of Fractions: Change any two of the numerator, denominator and symbol of the fraction itself, and the value of the fraction remains unchanged, that is, AAAA·BBBB.
Note: When applying the basic properties of fractions, we should pay attention to the limit condition C0 and the implied condition B0.
Fourth, the score of the score.
1. Definition: According to the basic properties of a fraction, reducing the common factor of the numerator and denominator of a fraction is called the reduction of the fraction.
2. Step: Factorize the denominator of the fractional numerator, and then remove the common denominator of the numerator and denominator.
3. Note: ① When the numerator and denominator of a fraction are monomial, the greatest common divisor of the coefficients of the fraction, numerator and denominator can be directly simplified, and then the lowest square of the same factor with the numerator and denominator can be simplified.
(2) If the numerator denominator is a polynomial, factorize the numerator denominator first, and then divide it.
4. simplest fraction's definition: When the numerator and denominator of a fraction have no common factor, it is called simplest fraction.
◆ About time sharing. Determination of common factor of numerator and denominator;
1) coefficient takes the greatest common divisor of the numerator and denominator coefficients as the coefficient of the common factor.
2) Take the lowest power of each common factor as the factor of the common factor.
3) If the numerator and denominator are polynomials, factorize the numerator and denominator first, and then judge the common factor.
V. General points of scores
1. Definition: A fraction with different denominators is converted into a fraction with the same denominator equal to the original fraction, which is called a general fraction of a fraction.
(foundation: the basic nature of the score! )
2. The simplest common denominator: take the product of the highest power of all factors of each denominator as the common denominator, and such a common denominator is called the simplest common denominator.
◆ How to determine the simplest common denominator in division:
1. coefficient takes the least common multiple of each denominator coefficient as the coefficient of the simplest common denominator.
2. Take the highest power of each common factor as the factor of the simplest common denominator.
3. If the denominator is polynomial, it is necessary to factorize each denominator first, and then judge the simplest common denominator.
Six, the four operations of the fraction and the power of the fraction
① Law of multiplication and division of fractions: the product of molecules is the numerator of the product, and the product of denominator is the denominator of the product. The formula is acacbbdbd.
Fraction divided by fraction: the numerator and denominator of division are reversed and then multiplied by divisor. The formula is acadad bdbcbc.
Ana② Power of Fraction: Power the numerator and denominator respectively. The formula is n bb.
③ Addition and subtraction of fractions: addition and subtraction of fractions with the same denominator: denominator unchanged, numerator addition and subtraction. The expression is: n
abab ccc
Addition and subtraction of fractions with different denominators: divide into fractions with the same denominator first, and then add and subtract. The formula is: acadbc bdbd.
Algebraic expression and fractional addition and subtraction: Algebraic expression can be regarded as an integer, with a negative sign in front of it and brackets, as a fraction with denominator of 1, and then divided by it.
(4) The operation sequence of the mixed operation of addition, subtraction, multiplication, division and multiplication of fractions.
Power first, then multiply and divide, then add and subtract. In the operation at the same level, whoever counts first and counts first with brackets should also pay attention to flexibility to improve the quality of problem solving.
Note: In the process of operation, it is necessary to make clear the purpose and basis of each step of deformation, and pay attention to the standard format of solving problems, which cannot be casually.
Skip to check the error or analyze the cause of the error.
The result of addition and subtraction must be reduced to the simplest fraction (or algebraic expression).
Seven. Integer exponential power
(1) After the introduction of negative integer and zero exponential power, the range of exponent is extended to all real numbers, and the method of positive integer power is adopted.
The same applies to negative integer exponential powers. Namely:
Amannam
NNN AMN ABAN BAN AMN AMN (A0)1anan0na0) a1(A0) (any number that is not equal to zero, zero abb
All powers are equal to 1)
Where m and n are integers.
Eight, the steps to solve the fractional equation:
(1) Remove the denominator and multiply both sides of the equation by the simplest common denominator of each denominator. (The process of generating additional roots)
⑵ Solve the whole equation and get the solution of the whole equation.
(3) Testing, substituting the solution of the obtained integral equation into the simplest common denominator:
If the simplest common denominator is 0, the original equation has no solution, and the value of this unknown is the root of the original equation; If the simplest common denominator is not 0, it is the solution of the original equation.
The conditions for increasing roots are: ① the solution of the integral equation obtained; (2) into the simplest common denominator, the value is 0.
Nine, fractional equation-basic steps:
(1) Review-Review the topic carefully and find out the equivalence relationship.
(2) Assumption-setting unknowns reasonably.
③ Column-list the equations (groups) according to the equivalence relation.
(4) solution-solve the equation (group). Attention test
5 Answer-Answer the question.