Exponent Problem!!!!?

Question

Which has greater value 75^50 or 50^75? Please explain your solution/proof...

Answer 1

75^50 or 50^75

75^50
75^(25 * 2)
(75^25)^2
((25 * 3)^25)^2
(25^25 * 3^25)^2
((5^2)^25)^2 * (3^25)^2
5^(2 * 25 * 2) * 3^(25 * 2)
5^100 * 3^50

50^75
50^(25 * 3)
(50^25)^3
((5^2 * 2)^(25))^3
((5^2)^25 * 2^(25))^3
(5^(2 * 25))^3 * 2^(25 * 3)
5^(2 * 25 * 3) * 2^(75)
5^(150) * 2^(75)
5^100 * 5^50 * 2^75
5^100 * (5 * 2)^50 * 2^25
5^100 * 10^50 * 2^25

since the 2 values have 5^100, you can take that out, so

3^50
5^50 * 2^75

so as you can see, since 75^50 has 3^50 and 5^50 * 2^75 as a difference between them

50^75 is much larger than 75^50

Sorry if it got confusing the way i worked it out.

Answer 2

50^75 has a greater value...

The first thing to do is to express both the exponents to a common base.

75^50 = (25 x 3)^50 = (5^2 x 3)^50
50^75 = (25 x 2)^75 = (5^2 x 2)^75

The first one is also equal to 5^100 x 3^50 while the second one is also equal to 5^150 x 2^75.

Note that 5^150 x 2^75 can also be written as 5^100 x 5^50 x 2^75. Since the first one is just equal to 5^100 x 3^50, then the latter expression or 50^75 is greater.

Answer 3

50^75= 26469779601696900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 OR 2.647E+127
75^50= 5663216564269380000000000000000000000000000000000000000000000000000000000000000000000000000000 OR 5.66322E+93
Thus 50^75 is greater than 75^50.
However do not go by the first response to your question.
Just consider the simplest problem of the type you mention.
What is greater 2^3 or 3^2?
2^3 =8
and
3^2=9
You will see that in this case the base is more weighty than power.

Answer 4

Power matters!

if the power is greater the final value would be greater.
hence, 75^50 < 50^75.

Answer 5

I would look at it this way:

75^50 = (50*1.5)^50 =50^50 *1.5^50
50^75 = 50^50 * 50^25

Divide out 50^50 from each, then square root 50 (i.e. 2root5) and set both exponents to 50 i.e.

75^50 = 50^50 * 1.5^50
50^75 = 50^50 * (2root5)^50

Which is bigger 1.5^50 or (2root5)^50? Well, both exponents being the same and both 1.5 and 2root5 being greater than 1, (2root5)^50 is going to be bigger than 1.5^50 because 2root5 (about 4.4721359549995793928183473374626) is bigger than 1.5 .

That's probably far from the simplest way of doing and it's probably not a proof but I think that clearly shows that for these particular numbers, 50^75 is the larger.

HTH

Answer 6

Answer is 50^75
because 75 times log50 is greater than 50 times log75

Answer 7

No matter how big the base number is (For example for 75^50,75 is the base),the power decides how big the number is.From here we can say that 50^75 is bigger although the base is smaller than that of 75^50.