How can you have a fraction as an exponent?

Question

P=250×2^(35/66)=361.05...

The question comes from a math book that doesn't give any reason how the 361 came about.
apparently 2 to the power of 0.53... times 250 equals 361.05... but I have no idea how that works. Does anyone here have a clue how an exponent can be a fraction? please don't confuse the question as a fraction with an exponent, I mean the exponent is a fraction.

Answer 1

As a simple example, the square root of 4 is also 4 to the 1/2 power. Written in root form, the denominator is outside of the root and the numerator is inside.

4^(1/2)=sqrt(4^1)=2

This is the same for any fraction.

4^(3/2)=sqrt(4^3)
=sqrt(64)=8

So 2^(35/66) means the sixty-sixth root of 2 raised to the thirty-fifth power.

Answer 2

Ok lets see here...normally when an exponent is a fraction it helps to put it in radical form as follows, If I were to take 16 and raise it to the 1/2 power, in radical form there would be a 2 in the V part of the radical and 16 would be underneath. Because the numerator in this case is 1, the 16 under the radical is not going to be raised to a power. But say we were take 16 and raise it to the 3/4 power. The denominator of the fraction always goes in the V part of the radical and the numerator goes underneath the radical and is the exponent for the number that is under it. So the 4 would be in the V part, the 16 would be under the radical and the 3 would be the power that the 16 would be raised to. So you would take 16 to the 3rd power which is 4096 then you would have the 4 still on the V part and under the radical you would have 4096. Then plug the numbers in the calculator and you would get 8.

So with your problem here 0.53 in fraction form is 53/100 so you would put the 100 in the V part of the radical and you would have 2 raised to the 53rd power under the radical. If you plug those numbers into a calculator then multiply it by 250 you would get 361 for an answer.

I hope this helps!

Answer 3

The power of 0.53 is the same as the power of 53/100 (0.53 is a decimal while 53/100 is its fractional equivalent). The numerator, 53, means that you take the 53rd power of the base (here, that is 2). The denominator, 100, means that you take the 100th root of that power.

A simpler example would be 25 to the 1/2 power. That means that we are looking for the square root of 25 (the denominator is 2). The square root of 25 is +/- 5

Answer 4

In your problem the 66 is the root and 35 is the power, but you would do this problem with logarithms. antilog(log 250 + 35/66*log 2)=
antilog(2.3979+ 35/66*.3010)= antilog 2.5575 = 361

Answer 5

if a fraction is an exponent it means its a square root. like, i.e. 25^1/2= 5
125^1/3= 5
etc.
savvy?