Middle school math puzzles, about square roots

1) Application of the cubic variance formula. a^3-b^3=(a-b)(a^2+ab+b^2)

The 4 cube root is 4 to the 1 3rd power of 4 i.e. 2 to the 2 3rd power, the 6 cube root is 6 to the 1 3rd power is 2 and 3 to the 1 3rd power multiply, the 9 cube root is 9 to the 1 3rd power that is 3 to the 2 3rd power, and then exponential.

1 1 (4 cube root + 6 cube root + 9 cube root) 3 (1 3)-2 (1 3).This is the cube root of 3 minus the cube root of 2.

2) The squared difference formula should be a 2-b 2 = (a + b) (a - b) 3) (2 + 3) + 2- 3) squared on both sides.

Re-open square, the value is 2 2

4)(x+y)^2

x^2+y^2+2xy

Re-open the square to get it, x+y= 5+1

There is no square root, it should be 2-3 is a negative number, and a negative number has no square root, let alone an arithmetic square root.

For junior high school, you don't know the conjugate virtual root, learn higher, imaginary numbers can also be squared to solve, but the level of knowledge understanding at different stages is different, some in junior high school are wrong in high school, some are wrong in high school, and some are wrong in high school, more precisely, the understanding is more and more complete, so I will not answer here.

c。2 , 3 , 2 - 3 0, and negative numbers have no square root, choose c

a, this is a negative number, there is no arithmetic square, the arithmetic square root is not a negative number, and negative numbers have two conjugate imaginary square roots.

This is a negative number, and later you will know that it has two conjugate square roots, and these two square roots are imaginary numbers, stipulating i*i=-1, but in junior high school, choose c.

How to open the root number in the root number: first calculate the small root number, and then open the root number again.

The first two parts of the question are about the simplification and calculation of the root number, and the middle parts are binary equations, that is, there are two unknowns, and the index is one calculation, and the latter is the simplification of the equations.

√18＝3√2

Simplification and the like:

3 2) ( 3 2) 3-2 = 1 (square difference formula).

5 3) ( 5 3) 5 + 3-2 15 = 8-2 15 (perfect square difference).

2 5 1) ( 5 1) ( 5 + 1) ( 5-1) + 5-1 = 5-1 + 5-1 = 3 + 5 (allocation rate, square difference).

4 5 1) (2 5 1) (2 5-1) +2 5 (2 5-1) = 10 + 1-4 5 + 10-2 5 = 21-6 5 (allocation rate, perfect square difference).

11 3) ( 11 3) 11-3 = 8 (square difference).

3x+4y=18 ①

5x-3y=1 ②

What are x,y respectively?

Solution: *3+ *4.

29x=58

x=2 to get y=3

x=2y=3y=3x+4 ①

3y-x=4 ②

x,y is?

Solution: *3- Gotcha.

x=-8 substitution.

y=-203x-4y=2 ①

x+3y=-1 ②

x,y is?

Solution: *3- Gotcha.

13y=-5

y=-5/13

Substitution. x=2/13

Factorization. Formula: one mention, two sets, three groups).

Mention is the mention of the common factor. Then the set of formulas (square difference, perfect square and cross multiplication) cannot be decomposed, so they are grouped and decomposed (sometimes the technique of adding and splitting terms is used).

x^+5x+6= (x+2)(x+3)

x^-6x+8＝ (x-2)(x-4)

x^-5x-24= (x-8)(x+3)

3x^-5x+2= (x-1)(3x-2)

8x^+18+7=(2x+1)(4x+7)

6x^-11x-10= (2x+5)(3x-2)

5x^-80= 5(x^-16)=5(x+4)(x-4)

Correct your mistake above first.

11 3) ( 11 3) 11-9=2 This part is a radical simplification.

3x+4y=18

5x-3y=1

What are x,y respectively?

The upper + lower formulas are:

8x+y=19, y=19-8x, substitute the following formula.

5x-3(19-8x)=1

x=2, so y=19-8*2=3

y=3x+4

3y-x=4

x,y is?

The upper formula is substituted for the lower formula.

3（3x+4）-x=4

x=-1, so y=3(-1)+4=1

3x-4y=2

x+3y=-1

x,y is?

The following formula is x=-1-3y, and it is substituted into the above formula.

3（-1-3y）-4y=2

y=-5 13,x=-1-3*(-5 13)=2 13 This part is a 2-element one-dimensional equation solution.

Simplification x +5x+6= (x+2)(x+3)x -6x+8 (x-2)(x-4).

x^-5x-24= (x-8)(x+3)

3x^-5x+2=(3x-2)(x-1)

8x^+18+7= (2x+1)(4x+7)6x^-11x-10= (2x-5)(3x+2)5x^-80= ..This one is already the easiest. 。What else do you want to do?

This part is factoring.

Well, each part is answered...

Since both radicals are equal to positive integers, the m minimum value is a = 3 5 = 15 and the n minimum value is b = 3 5 = 15, so a + b = 15 + 15 = 30

A positive number can open the square root, 2a-1=9 a=5

3a+b-1=16, from the above we know that the square root of a=5 b=2a+2b=5+2*2=9 9 is equal to plus or minus 3

Now that I have to use a calculator, why do I need to memorize any math tables?

Hehe, don't be angry.

In fact, I think that the first few commonly used ones can be memorized, and if the square root is memorized, I will memorize it here.

I don't know a cube root.

The number of squares can be from 1 to 20, and the number of cubes can be up to 11, which will not be a hard requirement, because if you don't use it much, you will forget it over time.

1 number written in the root number, 2 from the last digit that is the single digit forward to divide every two groups, the first remaining digit or two digits 3 for the first time divided according to the division rules, according to two points of one point, if there is one left, according to one digit, the remaining two digits are calculated according to two digits, leaving the remainder.

4 The second division, the divisor is 20 times the quotient + the quotient to be up, and the dividend is the remainder and the two digits pulled down, and each time it is pulled down two digits.

5 and so on, two digits and one point from the single place quantile, the first time except according to the division rules, the following divisor 20 multiply the quotient to quotient, until the division, or can not be divided, the zero is also two zeros.

Small case

In the same way, 8= 2*2*2= 2*2

Cannot be broken down. The root number should be decomposed first, and because of the attention symbol, the root number may be multiplied by 2 negative numbers, and the root number should be changed when the root number is opened.

Hope satisfied!

hundunlong．

√18＝3√2

Simplification and the like:

3 2) ( 3 2) 3-2 = 1 (square difference formula).

5 3) ( 5 3) 5 + 3-2 15 = 8-2 15 (perfect square difference).

2 5 1) ( 5 1) ( 5 + 1) ( 5-1) + 5-1 = 5-1 + 5-1 = 3 + 5 (allocation rate, square difference).

4 5 1) (2 5 1) (2 5-1) +2 5 (2 5-1) = 10 + 1-4 5 + 10-2 5 = 21-6 5 (allocation rate, perfect square difference).

11-3 = 8 (square difference).

3x+4y=18

5x-3y=1

What are x,y respectively?

Solution: *3+ *4.

29x=58

x=2 to get y=3

x=2y=3y=3x+4

3y-x=4

x,y is? Solution: *3- Gotcha.

x=-8 substitution.

y=-203x-4y=2

x+3y=-1

x,y is? Solution: *3- Gotcha.

13y=-5

y=-5/13

Substitution. x=2/13

Factorization. Formula: one mention, two sets, three groups).

Mention is the mention of the common factor. Then the set of formulas (square difference, perfect square and cross multiplication) cannot be decomposed, so they are grouped and decomposed (sometimes the technique of adding and splitting terms is used).

x^+5x+6=

x+2)(x+3)

x^-6x+8＝

x-2)(x-4)

x^-5x-24=

x-8)(x+3)

3x^-5x+2=

x-1)(3x-2)

8x^+18+7=(2x+1)(4x+7)

6x^-11x-10=

2x+5)(3x-2)

5x^-80=

5(x^-16)=5(x+4)(x-4)

The square root of arithmetic is positive: so it is: 3.

If a positive number has a square root, then there must be two, and they are opposites of each other. Obviously, if one of these two square roots is known, then the other square root can be obtained in time according to the concept of the opposite number.

Negative numbers cannot be squared in the real number system. It is only in the complex number system that negative numbers can be squared. The square root of a negative number is a pair of conjugate pure imaginary numbers. For example, the square root of -1 is i, and the square root of -9 is 3i, where i is an imaginary unit.

A positive number. If there are square roots, then there must be two, and they are opposites of each other. Obviously, if one of these two square roots is known, then the other square root can be obtained in time according to the concept of the opposite number. Negative numbers cannot be squared in the real number system.

It is only in the complex number system that negative numbers can be squared. The square root of a negative number is a pair of conjugate pure imaginary numbers. For example:

The square root of 1 is i, and the square root of -9 is 3i, where i is an imaginary unit.

√9=3

The arithmetic square root of 3 is 3

So: the arithmetic square root of root number 9 is root number 3.

Don't be fooled, answer 3.

9 is equal to 3, so the arithmetic square root of 3 is 3, thank you for adopting.