Question

SOS!!
In your own words explain the difference between the following two expressions:
(-1)^2/4 and (-1)^1/2

Answer 1

Negative one to the power of 2/4.
Negative one to the power of 1/2.
They both equal -1.5. I need the difference!

Answer 2

if we have:
\[(-1)^{\frac{ 2 }{ 4 }}\]
\[(-1)^{\frac{ 1 }{ 2 }}\]
Now there is a property of the exponents that allows me to convert the fractionary ecponents into one number, I will now prove it, but I will show it to you:
\[a ^{\frac{ m }{ n }}=\sqrt[n]{a ^{m}}\]
"a" is any number that is not zero, "m" and "n" are any real number but "n" cannot be zero.
So let's apply it to both cases and see what we get:
\[(-1)^{\frac{ 2 }{ 4 }}=\sqrt[4]{(-1)^{2}}\]
\[(-1)^{\frac{ 1 }{ 2 }}=\sqrt{(-1)}\]
This is the most important part, so we know that any negative number squared becomes positive, and the equare root of -1 is a complex number.
so we can write:
\[(-1)^{\frac{ 2 }{ 4 }}=\sqrt[4]{(-1)^{2}} = \sqrt[4]{1}=1\]
\[(-1)^{\frac{ 1 }{ 2 }}=\sqrt{(-1)} = i\]
so we can say that the difference between those two cases is that one gives us a real number and the other an imaginary number.

Answer 3

Wow thank you!!