Given that n is a positive integer and the 2n power of X is equal to 4, find the 2n power of (3N power of 3x) minus 13 (2n power of x)

Given that n is a positive integer and the 2n power of X is equal to 4, find the 2n power of (3N power of 3x) minus 13 (2n power of x)

X^2N=4
Original formula = 9x ^ 6n-13x ^ 4N
=9(X^2N)^3-13(X^2N)^2
=9 × 4^3-13 × 4^2
=368

Known: the third power of 1 = 1 = quarter × Square of 1 × Square of 2; The third power of 1 × The third power of 2 = 9 = quarter × Square of 2 square of 3; The third power of 1 + the third power of 2 + the third power of 3 = 36 = the square of quarter 3 and the square of 4; The third power of 1 + the third power of 2 + the third power of 3 + the third power of 4 = 100 = the square of quarter 4 and the square of 5 Please guess: what is the power of 1 + the power of 2 + the power of 3 +... + (n-1) and the power of N?

Square of quarter * n * (n + 1) = [n ^ 2 (n + 1) ^ 2] / 4

It is proved that the n-th power of 2 / a plus the n-th power of B is greater than or equal to the n-th power of 2 / A + B

Firstly, we should determine that a > 0, b > 0, n > = 2, n is an integer (a ^ n + B ^ n) / 2 > ((a + b) / 2) ^ n this problem can adopt the inductive method. When n = 1, the inequality is established. When n = k, (a ^ k + B ^ k) / 2 > = [(a + b) / 2] ^ k, the inequality is established. Both sides are multiplied by (a + B / 2) (a + B / 2) (a ^ k + B ^ k) / 2 > = ([(a + B / 2)] ^ (K + 1) [a

Given that the m-th power of 2 is equal to 1 / 16 and the n-th power of 1 / 3 = 27, find the value of the m-th power of n

The m-th power of 2 is equal to 1 / 16 = 2 ^ (- 4), and the n-th power of 1 / 3 = 27 = 3 ³
∴m=-4,n=-3
n^m=(-3)^(-4)=1/81

We know that because 4 is less than 5, the nth power of 4 is less than the nth power of 5 (n is a positive integer). Can you compare the size of the 100th power of 2 with the 75th power of 3? Hurry! Hurry!

2^100=4^50=8^25
3^75=9^25
Since 9 ^ 25 is greater than 8 ^ 25, 3 ^ 75 is greater than 2 ^ 100

If a > b > 0 and N is a positive integer, the n-th power of a is greater than the n-th power of B. use this knowledge to compare the size of the 100th power of 2 with the 75th power of 3

2^100=(2^4)^25=16^25
3^75=(3^3)^25=27^25
∵16＜25
∴16^26＜27^25
∴2^100＜3^75