The nature of the quadratic radical, what is the quadratic radical nature?

The application of the quadratic radical formula is mainly reflected in two aspects: the use of important ideas and methods from special to general, and then from general to special, to solve some exploratory problems of laws; The quadratic radical formula is used to solve the problem of length and height calculation, and some length or height is obtained according to the known quantity, or the scheme of material saving is designed, as well as the splicing and segmentation of the figure. This process requires the use of quadratic root calculations, which is actually simplification of evaluation.

Is there a difference between a quadratic root and an arithmetic square root?

The first and second radicals are an algebraic formula, while the arithmetic flat root is an operation.

Second, the quadratic radical formula is richer than the square root of arithmetic.

3. Quadratic radicals must have a root number, while arithmetic square roots do not necessarily have a root number.

Fourth, quadratic radicals can be regarded as arithmetic square roots, and the arithmetic square roots represented by root numbers are also quadratic radicals.

i.Definition and Concept of Quadratic Radicals: 1. Definition: In general, an algebraic expression of the form ā(a 0) is called a quadratic radical. When a 0, a is the square root of the arithmetic of a, 0=0

2. Concept: The formula ā(a 0) is called a quadratic radical. āa 0) is a non-negative number. ii.The simple properties and geometric significance of the quadratic radical ā in this paragraph 1)a 0 ; 0 [ Double non-negativity ]

2) ( 2=a (a 0) [any non-negative number can be written as the form of a number squared].

3) a 2 + b 2) denotes the distance between two points between planes, i.e., the Pythagorean theorem inference. iii.The nature of the quadratic radical and the simplest quadratic radical This paragraph 1) Simplification of the quadratic radical ā.

a(a≥0）

|a|={a(a＜0)

2) The square root of the product and the square root of the quotient.

ab=√a·√b（a≥0，b≥0）

a/b=√a /√b（a≥0，b>0）

3) The simplest quadratic radical.

Conditions: 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 or factors that can be converted into squares or squares.

For example, there are 2, 3, a(a 0), x+y, etc., which do not contain factors or factors that can be squared or squared;

Factors or factors that can be squared or squared are 4, 9, a 2, (x+y) 2, x 2+2xy+y 2, etc. Satisfied.

Properties of Quadratic Radicals:1. A denotes the arithmetic square root of a.

According to the non-negativity of the arithmetic square root, the quadratic root a(a 0) is a non-negative number.

2. Quadratic radical A 2 LAL. This nature can be divided into three situations.

3. The arithmetic square root property of the quadratic radical disturbance product: ab a b (a 0, b 0).

4. The arithmetic square root property of the quadratic radical quotient: a b a b (a 0, b 0).

The simplest two-leakage bending crack secondary root type:1. There is no denominator in the number of squares to be opened.

2. The number of squares to be opened does not contain factors and factors that can be opened to the end.

The property of the arithmetic square root of the product: The arithmetic square root of the product is equal to the product of the arithmetic square root of each factor in the product.

The properties of quadratic radicals are as follows:

1. There are two square roots of any positive number, and they are opposite to each other. Such as the arithmetic square root of a positive number.

is a, then the other high of a is changed to a square root of a,; In the simplest form, the number of squares to be opened cannot have a denominator.

Exist. 2. The square root of zero is zero.

3. There are also two square roots of negative numbers, which are conjugated.

4. Rationalized radicals: If there are two algebraic formulas containing radicals.

The product of no longer contains an radical, then these two algebraic formulas are rationalized radicals of each other, also known as rationalized factors.

Addition, subtraction, multiplication and division of quadratic radicals

Addition and subtraction of the first and second radicals.

1. The same kind of quadratic radical type.

In general, a few quadratic radicals are reduced to the simplest quadratic radicals.

Finally, if they have the same number of squares, these quadratic radicals are called the same quadratic radicals.

2. Merging the same quadratic radicals: Merging several quadratic radicals of the same kind into one quadratic radical is called merging the quadratic radicals of the same kind.

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

2. Multiplication and division of quadratic radicals.

Quadratic radical multiplication and division, multiply and divide the open square, the root exponent remains unchanged, and then turn the result into the simplest quadratic radical.

1. Multiplication operation: The product of the arithmetic square root of two numbers is equal to the arithmetic square root of the factor product of these two dusts.

2. Division: The quotient of the arithmetic square root of two numbers is equal to the arithmetic square root of the quotient of these two numbers.

Modification of the concept and properties of quadratic radicals:

In general, a formula of the shape a(a 0) is called a quadratic radical, where " " is called the quadratic root number and a is called the square number to be opened.

For example, 2, (x 2+1), x-1) (x 1), etc. are all quadratic radicals.

Properties of Quadratic Radicals:

When a 0, a denotes the arithmetic square root of a, so a is a non-negative number (a 0), i.e., for equation a, not only a 0, but also a 0, so it can be said that a has double non-negativity.

Minimal quadratic radical:

1. There is no denominator in the number of squares to be opened.

2. The number of the opened party does not contain the factors and factors that can be opened to the end.

The property of the arithmetic square root of the product:

The arithmetic square root of the product is equal to the product of the arithmetic square roots of the factors in the product.

The property of the arithmetic square root of the quotient:

The arithmetic square root of the quotient is equal to the arithmetic square of the divided** arithmetic square root of the dividing formula.

Note: The deformation of the root number in the denominator is called the rationalization of the denominator, and the method of rationalizing the denominator is to multiply the numerator and the denominator by the rationalized factor of the denominator respectively according to the basic properties of the fraction (multiplying two algebraic formulas containing quadratic radicals, if their product does not contain quadratic radicals, it is said that these two algebraic formulas are rationalized factors for each other) to remove the root number in the denominator.

1.There are two square roots of any positive number, which are opposites of each other. For example, the arithmetic square root of a positive number a is.

Then the other square root of a is

In the simplest form, there can be no denominator for the number of squares to be opened.

2.The square root of zero is zero, ie.

3.There are also two square roots of negative numbers, which are conjugated. For example, the square root of a negative number a is.

4.Rationalized radicals: If the product of two algebraic formulas containing radicals no longer contains radicals, then the two algebraic formulas are rationalized with each other, also known as rationalized and chemical factors with each other.

5.Irrational numbers can be expressed in the form of continuous fractions, such as:

6.When a 0, > "and <>

The range of values in a is the entire blind letter complex plane.

Any number can be written in the form of a number squared; This property allows for factorization.

8.Inverse moves non-negative factors outside the root number into parentheses, such as.

a>0） ，a<0），>a≥0_ ，a<0）。

9.Note: > then removes the absolute value sign based on the operation of the absolute value.

10.It has a double non-negativity, i.e. not only a 0 but.

Definition of quadratic radicals: In general, we call the algebraic formula of the form a(a 0) a quadratic radical, where a under the root sign is called the square number. When a 0, a is meaningful, representing the arithmetic square root of a; When a is less than 0, a is meaningless.

Properties of quadratic radicals: (1) a(a 0) is a non-negative number, i.e. 0(a 0).

2) The number of the square of a is a non-negative number, i.e. a 0.

Chanchang (3)( a) = a(a 0), i.e., the square of the arithmetic square root of a non-negative number is equal to itself.

4)√a=|a|(a 0), i.e., the arithmetic square root of any number is equal to the absolute value of that number.

Nature 1The square root of any positive number bai has two du, which are opposite to each other. If the arithmetic square root of the positive number a is x, then the other square root of a is x.

2.The square root of the answer to zero is zero, ie.

3.Negative numbers do not have square roots.

4.Rationalized radicals: If the product of two algebraic formulas containing radicals no longer contains radicals, then the two algebraic formulas are rationalized radicals of each other, also known as rationalized factors.

5.Irrational numbers can be expressed in the form of rational numbers

The sexual purification of the quadratic radical form is:

1) Sakura A 0 (A Spine Shirt 0);

2)(√a)^2=a(a≥0);

3)√(a^2)=|a|=a(a≥0)

a^2)=|a|==a(a0).