It is proved that (1) the product of two n-th unit roots is still a n-th unit root (2) The reciprocal of an n-th unit root is still an n-th unit root (3) All n-th power roots of complex Z can be obtained by multiplying one n-th power root of Z by all n-th power unit roots

It is proved that (1) the product of two n-th unit roots is still a n-th unit root (2) The reciprocal of an n-th unit root is still an n-th unit root (3) All n-th power roots of complex Z can be obtained by multiplying one n-th power root of Z by all n-th power unit roots

If clearly defined, these conclusions can be verified directly
By definition, the complex number a is a unit root of degree n if and only if a ^ n = 1
(1) If a and B are n-th unit roots, then a ^ n = B ^ n = 1
So (AB) ^ n = a ^ n · B ^ n = 1, that is, AB is also the unit root of degree n
(2) If a is the unit root of degree n, then a ^ n = 1
Obviously, a ≠ 0, 1 / A has a definition, and (1 / a) ^ n = 1 / A ^ n = 1, that is, 1 / A is also the root of n-th unit
(3) First of all, if z = 0, then the n-th root of Z is only 0, and the proposition is obviously true. The following only considers the case of Z ≠ 0
If a is an nth root of Z, then a ^ n = Z
For any nth root B of Z, B ^ n = Z. then (B / a) ^ n = B ^ n / A ^ n = 1
That is, B / A is a unit root of degree n, so B = a · (B / a) can be written as the product of a and a unit root of degree n
On the contrary, if C is a unit root of degree n, then C ^ n = 1
So (AC) ^ n = a ^ n · C ^ n = Z, that is, AC must be the nth root of Z

If n is even and N ∈ n +, a + b > 0, we prove that B ^ (n-1) / A ^ n + A ^ n-1 / b ^ n ≥ 1 / A + 1 / b

b^(n-1)/a^n + a^(n-1)/b^n - 1/a - 1/b≥0
b^(n-1)/a^n - 1/a=[b^(n-1)-a^(n-1)]/a^n
[b^(n-1)-a^(n-1)]/a^n - [b^(n-1)-a^(n-1)]/b^n≥0
[b^(n-1)-a^(n-1)] ( 1/a^n - 1/b^n )≥0
Let b) ≥ a, then B ^ (n-1) > = a ^ (n-1)
Because a + b > 0
Then b > 0, | B |) ≥ | a|
And N is even
b^n≥a^n>=0
1/a^n - 1/b^n)≥0
Proof of the original formula

The fruit shop delivered a batch of apples. Two fifths of the total apples were sold on the first day, and the remaining five fifths were sold on the second day. There were 180 kilograms left. How many apples were there

Suppose that the quality of the apples is X
（1-2/5）x*3/5=180
The solution is x = 500kg

One half +. + ninety nine percent + one hundred and one hundred percent

1 / 2 + 2 / 3 + 3 / 4 + 4 / 5 +. + 99 / 100 + 100 / 101 = 1-1 / 2 + 1-1 / 3 + 1-1 / 4 + 1-1 / 5 +. + 1-1 / 100 + 1-1 / 101 = 100 - (1 / 2 + 1 / 3 + 1 / 4 + 1 / 5 +. + 1 / 100 + 1 / 101) this is a harmonic series, there is no general formula, there is an approximate formula 1 + 1 / 2 + 1 / 3 + +1 / N = lnn ln is a natural logarithm, when n tends to infinity, 1 +..., 1 +..., 1 +..., 1 +..., 1 +..., 1 +..., 1 +..., 1 +..., 1 +..., 1 +..., 1 +..., 1 +..., 1 +..., 1 +

Fundamentals of circuit analysis: how to judge the capacitance and sensibility of a circuit
 

Because US = 10V, ur = 6V, ul-uc = 8V,
Because UL = 4V, UC = 12V,
It can be concluded that XC > XL, the circuit is capacitive

The number of fifth grade students is between 40 and 50. If they are divided into a group of eight, there will be five more students in one group. If they are divided into a group of 12, there will be three groups each
The number of fifth grade students is between 40 and 50. If they are divided into a group of eight, there will be five more students in one group. If they are divided into a group of 12, there will be one less student in each of the three groups,

According to the meaning of the question, three people are missing in both methods,
Find the common multiple of 8 and 12, then subtract 3
The common multiples of 8 and 12 are: 24 48, 48-3 = 45
There are 45 students in grade five

Is a symmetric matrix a positive definite matrix? How to prove it?

No
as
-1 0
0 1

Factorization of x ^ 4 + x ^ 3 + x ^ 2 + X + 1 in the range of real and complex numbers
Questions of self enrollment of Shandong University 08

Original formula = (x ^ 5-1) / (x-1)
First, find out the root of x ^ 5-1 = 0, and then remove the root of 1
From x ^ 5-1 = 0, we know that x is the root of unit circle of degree 5,
So X1 = 1, X2 = cosa + Sinai, X3 = cos2a + sin2ai, X4 = cos3a + sin3ai, X5 = cos4a + sin4ai
Where a = 2 Π / 5
So the original formula = (x-x2) (x-x5) (x-x3) (x-x4)

There are 46 students rowing, a total of 10 boats (each boat is full), including 6 people for each big boat, 4 people for each small boat, the big boat is full______ Article

Suppose there are x large boats, then there are (10-x) small boats, 6x + 4 × (10-x) = 46, & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; 6x + 40-4x = 46, & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; 2x = 6, & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; & nbsp; X = 3

lim（2x+1）sinx/2/x+2=

LIM (2x + 1) SiNx / 2 / x + 2 the last + 2 is not the denominator? X tends to 0
=lim（2x+4-2）sinx/2/x+2
=2-2limsinx/2 / x+2
=4-limsinx/2 / x/2
=4-1
=3