89^90 and 90^89....which is greater.please give me the mathematical proof?

Question

I want a mathematical proof.....

Answer 1

math_kp partially solved a similar problem last night:

Take the derivative of the function y = x ^ (1/x),
y' = y / x^2 * (1 - ln(x))
and notice it's 0 at x = e, and negative for x > e.

Since a = 89 and b=90, both a,b > e, and a a^(1/a) > b^(1/b) or
a^b > b^a
89^90 > 90^89

Answer 2

A-B =C
if C>0 than A > B
if C<0 than A
using calculator
89^90 - 90^89 =X , X>0
90^89-89^90 =X , X<0

so conclusion
89^90 > 90^89

Answer 3

You dont need mathematical proof - they are just numbers.

89^90 = 2.78676661 × 10^175

90^89 = 8.46414978 × 10^173

So 89^90 is greater by a factor ~ 100

Answer 4

I have proved that x^(1/x) decrease with x increasing for x >e
please see the link below
so 89^(1/89)> 90^(1/90)

raise them to power 89 * 90

89^90 > 90^89

Answer 5

you can look at how it works with smaller numbers and see if there is a trend.

so look at
1^2 and 2^1
2^3 and 3^2
3^4 and 4^3
3^5 and 5^4

we get
1<2
8<9
81>64
1,024>625
15,625>7,776

you will see when you look at this that though the first 2 the secong formula is bigger as the numbers get bigger the 1st number is bigger than the second one.

This trend will follow hence 89^90 > 90^89

Answer 6

89^90 is 89*89*89... 90 times
while
90^89 is 90*90*90... 89 times... I guess they would be close as numbers go... so let's work it out

excel gives
2.7868E+175
8.4641E+173

so the first is greater..... 89^90 > 90^89

Answer 7

The rote way is to just use the calculator
to find the values and see which is greater.
We will try something different.
Assume 89^90 is greater than 90^89
Taking logs we have
90 log89 gr 89 log 90
90/89 gr log90/log89
1.011 gr1.954/1.949
1.011gr1.002
What we assumed is true
Hence proved

Answer 8

simple proof

89^90 >? 90^89 ... take log
90 log 89 >? 89 log 90
(90/89) >? (log 90 / log 89) .... this is true, thus proven

Answer 9

89^90=(89^2)^45
=8189^45
=(8189^44)*8189 ok.
now take 90^89=(90^88)*90=(8100^44)*90. now since 8189>8100and 8189>90 we can conclude that 89^90>90^89.
thanx maths kp. my answer sucks. 89 squared is 7921 and not 8189. so my proof is invalid. thinking of another solution.