Given a & gt; 0, a is not equal to 1, M & gt; n & gt; 0, let a = a ^ m + A ^ - m, B = a ^ n + A ^ - N, try to compare the size of a and B

Given a & gt; 0, a is not equal to 1, M & gt; n & gt; 0, let a = a ^ m + A ^ - m, B = a ^ n + A ^ - N, try to compare the size of a and B

A-B = a ^ M-A ^ n + 1 / A ^ M-1 / A ^ n general = (a ^ 2m * a ^ n-a ^ m * a ^ 2m + A ^ n-a ^ m) / A ^ m * a ^ n obviously denominator a ^ m * a ^ n > 0 numerator = a ^ 2m * a ^ n-a ^ m * a ^ 2m + A ^ n-a ^ m = a ^ m * a ^ n (a ^ M-A ^ n) - (a ^ M-A ^ n) = (a ^ M-A ^ n) (a ^ m * a ^ n-1) if 0A ^ 0 = 1, similarly a ^ n > 1, so a ^ m * a ^ n > 1, a ^ m * a ^ n-1 >

If M is greater than N, n is greater than 0, a is greater than 0, and a is not equal to 1, try to compare the size of (m times of a + m times of a) and (n times of a + n times of a)
How to compare? What is the idea?

The product of two addends in the formula to be compared is equal (both are 1), so the comparison can be converted into two groups of positive numbers whose products are equal. The comparison and conclusion is that the greater the difference is, the greater the sum is. It is proved that if AB = CD, a > C > d > b > 0, then A-B > C-D > 0 (a + b) ^ 2 = (a-b) ^ 2 + 4AB = (a-b) ^ 2 + 4CD > (C-D) ^ 2 + 4CD = (c + D)

Given that M = a ^ 3B ^ 2, n = a ^ 2B ^ 3, AB is not equal to 0, and a > b, try to compare the size of M and n

M-N = a ^ 3B ^ 2-A ^ 2B ^ 3 = a ^ 2B ^ 2 (a-b) > 0, so m > n

Let a not equal to 2, B not equal to 1m = the square of a + the square of B, n = 4a-2b-5, try to compare the size of M and n

M-N
=a²+b²-4a+2b+5
=a²-4a+4+b²+2b+1
=(a-2)²+(b+1)²
a≠2 (a-2)²>0
(B + 1) & sup 2; constant nonnegative
(a-2)²+(b+1)²>0
M>N

If a > 0, b > 0, a + B = 1, then the minimum value of AB + 1 / ab

a>0,b>0
1= a+b>=2√(ab)
√(ab)

Y = sin ^ 3x, for the n-th derivative of Y, I know it's using Leibniz formula, and it's better to give the answer

The result is complex and can not be simplified. In fact, the method is to apply Leibniz formula twice. See the attached figure for details

13/80*3/7*40
Division is the best way to score fast now
Simple calculation

13/80×3/7×40
=13/20×3/7
=39/140

What is the difference between the quotient of 107 divided by 2 and the reciprocal of 7? ② 34 of a is equal to 45 of B. if B is 15, what is a?

① (107 △ 2) - 1 △ 7 = 57-17, = 47; a: the difference is 47. ② 15 × 45 △ 34 = 12 × 43. = 16

4X of 3-2x of 5 = 7 of 8 2x of 3 / 1 of 4 = 12

4x/3-2x/5=7/8
20x/15-6x/15=7/8
14x/15=7/8
x=15/16
(2x/3)/(1/4)=12
2x/3= 1/4× 12=3
x=9/2

When 57 × 101 is calculated by a simple method, it is based on
A: Additive commutative law B, multiplicative commutative law C, multiplicative distributive law D, multiplicative associative law

C. Multiplicative distributive law
57×57=57×（100+1）
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