Mathematical cube roots will involve many aspects, so what knowledge points should we master? The following is a summary of the knowledge points of the mathematical cube root that I recommend to you, hoping to bring you help.

Summarize the knowledge points of mathematical cube root: if the cube of a number X is equal to A, that is, the cube of X is equal to A (X 3 = A), that is, X is equal to A three times in a row, then this number X is called the cube root of A.

cube root

Reading? Cube root a? Where a is called the root number and 3 is called the root index. (A equals all numbers, including 0) If the root number is an exponent, then this exponent (which must be divisible by three) can also be divisible by the cube root number.

The operation of finding the cube root of a number is called root extraction.

The nature of the cube root:

The cube root of (1) positive numbers is positive numbers. (2) The cube root of a negative number is a negative number. (3) The cube root of 0 is 0. Generally speaking, if the cube of a number X is equal to A, then this number X is called the cube root of A (also called cube root). For example, 2 is the cube root of 8, -2/3 is the cube root of -27/8, and 0 is the cube root of 0.

Cubes and squares are reciprocal operations.

The cube roots of two opposing numbers are also opposing.

Negative numbers cannot be squared, but they can be opened.

How does the cube root compare with other figures? (1) Do the cube of these two numbers.

(2) Work

(3) Compare the number of roots (e.g. cube root number 3 is greater than cube root number 2)

Any number (positive, negative or zero) can only have one cube root, if it exists.

Difference and connection between square root and cubic root

First of all, the difference

⑴ The root index is different: the root index of the square root is 2 and can be omitted; The root index of the cube root is 3 and cannot be omitted.

⑵ The range of roots is different: the number of roots must be non-negative; The number of roots in a cube can be any number.

⑶ Different results: the square root has two opposite results except 0; The cube root has only one result.

Second, contact

Both are contrary to the operation of power.

Knowledge point one:

The concept of square root: If x2=a(a? 0), then x is called the square root of a and recorded as x=? The operation of finding the square root of a non-negative number is called square root.

Example 1

The square root of is ().

A.？ 9 B？ 3 C.9 D.3

Solution: Because

=9, so

The square root of is the square root of 9, that is?

=? 3, so choose B.

Note: we should now

Simplify and then evaluate.

Knowledge point 2:

The concept of arithmetic square root: the positive square root of a positive number is called the arithmetic square root of a, and the arithmetic square root of 0 is 0.

Example 2 If one

A.- Bachelor's degree? A d?

Solution: When

=|a|=-a, so choose a.

Example 3 The arithmetic square root of a number is a, so the number that is 5 larger than this number is ().

A.a+5 B.a-5 C. a2+5 D. a2-5

Solution: If the arithmetic square root of a number is A, then this number is a2, and the number 5 greater than this number is a2+5, so choose C.

Knowledge point 3:

Properties of square root and arithmetic square root: 1. Positive numbers have two square roots, and the two square roots are in opposite directions; The square root of 2.0 is 0; 3. Negative numbers have no square root; 4. the arithmetic square root of non-negative numbers is non-negative, that is, a? 0.

If the square roots of m are 2a-3 and a- 12, find the value of m 。

Solution: A positive number has two square roots, and the two square roots are in opposite directions. (2a-3)+(a- 12)=0, the solution is a=5, so m=(2a-3)2=72=49.

Example 5 If 2a-3 and a- 12 are the square roots of m, find the value of.

Analysis: This example seems to be the same as Example 4, but it is not. Because "if the square roots of m are 2a-3 and a- 12", we know that 2a-3 and a- 12 are opposites, and "if 2a-3 and a- 12 are square roots of m", we can know that 2a-3 and a are opposites. A= -9。 So 2a-3=- 18-3=-2 1, so m=(-2 1)2=44 1. (2) When (2a-3)+(A-65438+)

Knowledge point 4:

Concept and properties of cube root: If x3=a, then X is called the cube root of a, and the cube root marked as x=.0 is 0. Any real number has a cube root, and there is only one cube root. The symbol of the cube root is the same as its own symbol.

Knowledge point 5:

Find the square root, cube root, etc. with a calculator.

Example 8 (Shaanxi Province) Compare dimensions with a calculator:

(Fill in ">", "=" and "<).

Analysis: this kind of question is to examine the process of students using the calculator and pay attention to the order of keys. Therefore, fill in >; .

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