How do you do addition and subtraction operations with quadratic radicals?

Strategies and methods for simplification and computation of quadratic radicals.

Quadratic radicals are a difficult content in junior high school mathematics teaching, and readers generally follow the following practices when simplifying and operating quadratic radicals after mastering the concepts and properties related to quadratic radicals

First, the quadratic radical in the equation is appropriately simplified.

The multiplication of quadratic radicals can be carried out with reference to polynomial multiplication, and formulas are used in the operation.

For the division of quadratic radicals, the form of the component formula is usually written first, and then the operation is carried out by rationalization of the denominator

The addition and subtraction of quadratic radicals is similar to the addition and subtraction of polynomials, that is, the parentheses are removed and similar terms are merged on the basis of simplification

The result of the operation is generally reduced to the simplest quadratic radical

Common techniques and methods for simplifying quadratic radicals.

The simplification of quadratic radicals is an important part of the teaching of quadratic radicals, and for the simplification of quadratic radicals, in addition to mastering the basic concepts and algorithms, it is also necessary to master some special methods and skills, which will get twice the result with half the effort.

The addition and subtraction of quadratic radicals is as follows:

Knowledge point 1: Quadratic radicals of the same kind.

After several quadratic radicals are transformed into the simplest quadratic radicals, if the number of squares is the same, the first beat of these quadratic radicals is called the same kind of quadratic radicals, such as such quadratic radicals are all the same kind of quadratic radicals.

The method of judging the quadratic radical of the same kind:

1) First of all, after the quadratic radical that is not the simplest form is converted into the simplest quadratic radical, then see if the number of squares is the same.

2) Whether several quadratic radicals are the same kind of quadratic radicals is only related to the number of squares opened and the root exponent, and has nothing to do with the dialectic of the other than the root number.

Knowledge point 2: The method of merging the same kind of quadratic radicals.

The theoretical basis for merging the same quadratic radicals is the distributive law of inverse multiplication to addition, merging the same quadratic radicals, only adding their coefficients, the root exponent and the number of open squares are unchanged, and non-homogeneous quadratic radicals cannot be combined.

Knowledge point 3: The law of addition and subtraction of quadratic radicals.

The addition and subtraction of quadratic radicals first turns each quadratic radical into the simplest quadratic radical, and then merges the same quadratic radicals, and the method of merging is the addition of coefficients, and the radicals remain unchanged.

Knowledge point 4: Hybrid operation methods and sequences of quadratic radicals.

The method of operation is a mixture of addition, subtraction, multiplication, division, and rules similar to polynomial multiplication. The order of operation is multiplication first, then multiplication and division, and finally addition and subtraction.

Knowledge point 5: The difference between the law of addition and subtraction of quadratic radicals and the law of multiplication and division.

In multiplication and division, the coefficients are multiplied and the number of squares is multiplied, regardless of whether the two formulas are the same radicals, and in addition and subtraction, the coefficients are added, the number of squares is unchanged and the two squares must be the simplest radicals of the same kind.

The method of merging is similar to that of merging similar terms, adding the factors (formulas) outside the root number, and the root exponent and the number of open squares remain unchanged. To determine whether two quadratic radicals can be merged in addition and subtraction, it must be reduced to the simplest quadratic radical first, and then see if the number of open squares is the same, if the same, they can be merged, otherwise they cannot be merged.

For the addition and subtraction of quadratic radicals, we can first talk about the quadratic radical into the simplest quadratic radical, find the quadratic radical with the same number of opened squares after simplification, and finally merge the quadratic radicals with the same number of opened squares and add the coefficients as coefficients, and the root coefficient and the number of opened squares remain unchanged.

The essence of the addition and subtraction of the quadratic radical is to merge the same quadratic radicals, that is, first make each quadratic radical into the simplest quadratic radical, and then merge the same quadratic radicals, and for the quadratic radicals that are not merged, they still need to be written into the result.

In the addition and subtraction of quadratic radicals, the commutative and associative properties and the rules of removing brackets and adding brackets in the integer addition and subtraction operations are still applicable. The mixed order of operation of the quadratic radical is the same as the order of operation in real numbers, first multiplication, then multiplication and division, and finally addition and subtraction. The arithmetic and multiplication formulas in real and integer operations are still applicable in quadratic radical operations.

Get 0. In general, an algebraic formula of the shape a is called a quadratic radical, where a is called the square number. When a 0, a is the arithmetic square root of a; When a is less than 0, the value of a is a pure imaginary number (in the formula for finding the roots of a quadratic equation, if the number under the root sign is negative, then the equation has two conjugate imaginary roots).

The main way to judge whether a quadratic radical is the simplest quadratic radical is to follow the definition of the simplest quadratic radical, or intuitively observe that the exponent of each factor (or factor) of the open square is less than the root exponent 2, and the open square does not contain a denominator.

If the square of a number is equal to a, then the number is called the square root of a. A can be a concrete number or an algebraic formula containing letters.

i.e.: if, then.

It is called the square root of a and is denoted as x=

where a is called the number of squares to be opened. The square root of the positive is called the arithmetic square root. Regarding the concept of quadratic radicals, it should be noted that:

The square number can be either a number or an algebraic formula. If the square number is positive or 0, its square root is a real number; If the number of squares to be opened is negative, its square root is an imaginary number.

In general, an algebraic formula like a is called a quadratic radical, and the quadratic radical addition and subtraction calculation must first convert the quadratic radical to the simplest quadratic radical, and then merge the same number of squares.

In general, an algebraic formula of the shape a is called a quadratic radical, where a is called the square number. When a 0, a is the arithmetic square root of a; When a is less than 0, the value of a is a pure imaginary number (in the formula for finding the roots of a quadratic equation, if the number under the root sign is negative, then the equation has two conjugate imaginary roots). The main way to judge whether a quadratic radical is the simplest quadratic radical is to follow the definition of the simplest quadratic radical, or intuitively observe that the exponent of each factor (or factor) of the open square is less than the root exponent 2, and the open square does not contain a denominator.

1.Homogeneous quadratic radical.

Generally speaking, after several quadratic radicals are reduced to the simplest quadratic radicals, if they have the same number of squares, these quadratic radicals are called the same quadratic radicals. Simplification: Root number 12 is equal to root number 3 of 4

2.Merge the same quadratic radicals.

Merging several homogeneous quadratic radicals into one quadratic radical is called merging homogeneous quadratic radicals.

3.When adding or subtracting quadratic radicals, you can first convert the quadratic radicals to the simplest quadratic radicals, and then merge the same number of squares.

1.Turn fractions or decimals into false fractions.

2.Factoring the square number into prime factors or factoring.

3.Move the factors or factors that can be opened as far as possible in the root sign outside the root number.

4.Remove the denominator in the root number, or remove the denominator with the sliding root number.

5.Approximation. 1.The factor of the open square number is an integer or letter, and the factor is an integer.

2.The number of squares to be opened does not contain factors that can be turned into squares or squares, or because of the old town of jujube.

Quadratic Argument Radical:

Addition and subtraction: only the number of squares being opened (i.e.

7 in 7) can only be added or subtracted.

Example 1（1）

Multiplication and division: Example 2

That is, multiply the number of squares to be opened).

2)2 3 posture foci 2

The result should be the simplest root (i.e., no denominator and no factor that can be broken down into integers).

Here's an example:

Addition and subtraction of quadratic radicals.

1 Homogeneous quadratic radicals Generally speaking, after several quadratic radicals have been reduced to the simplest quadratic radicals, if they have the same number of squares, these quadratic radicals are called homogeneous quadratic radicals. 2 Merging Homogeneous Quadratic Radicals Merging several homogeneous quadratic radicals into one quadratic radical is called merging homogeneous quadratic radicals. 3. When adding or subtracting quadratic radicals, you can first convert the quadratic radicals into the simplest quadratic radicals, and then merge the same number of squares.

For example: 2 5 + 5 = 3 5

8 + root number 18 = 8 + 3 times root number 12

First approximate 3, then multiply by the root number A-2, and subtract 2a from the root number A of A-2

For example, what is the root number 8 + root number 18?

Answer: Original formula = 2 times the root number 2 plus 3 times the root number 2 = 5 times the root number 2

(1) Root number 8 + root number 18

2 times the root number 2 + 3 times the root number 2

5 times the root number 22) 3 times the root number A - 3 times the root number A of the 2nd time

A - 2/A under the root number

A-2 times the root number A-1

The addition of the quadratic radical.

BAI subtraction rule.

When the secondary root du formula is added or subtracted, it can be.

zhi first converts the second DAO sub-radical version into the simplest quadratic radical, and then merges the quadratic radicals with the same prescribe weight

Note: (1) The addition and subtraction of quadratic radicals is usually divided into two steps, the first step is simplified; Step 2: Merge;

2) Before merging, it should be noted that it is necessary to determine which of the quadratic radicals of them have the same number of open squares; When merging, it is similar to the previous merge of similar terms, only the factor outside the root number needs to be added or subtracted, and the number of squares and the root exponent remain unchanged

Simplify and add or subtract, or use a calculator to calculate directly.