What is (-1)^(3/5) and (-1)^(6/10)?

Question

anonymous · Answer

First, since 6/10 is reducible to 3/5, both the given expressions evaluate to the same value. To evaluate the expression, break down the exponent into two parts ^3 and ^(1/5). The first part is -1^3, which equals -1 * -1 * -1 = -1. The second part is -1^(1/5) = the fifth root of -1.

Can you think of a number, which when multiplied with itself five times, gives you -1? That is the answer

anonymous · Answer

Can i write (-1)^(6/10) as( (-1)^6)^(1/10) which is 1?

solomonzelman · Answer

$$ (-1)^{3/5}~~~and~~(-1)^{6/10}$$same as $$ \sqrt[5]{(-1)^3}~~~and~~ \sqrt[10]{(-1)^6}$$which gives$$ \sqrt[5]{-1}~~~and~~ \sqrt[10]{1}$$$$ -1~~~and~~ 1$$

solomonzelman · Answer

Whatever "and" is supposed to mean here.

anonymous · Answer

@SolomonZelman I think it is (-1)^(3/5)

kainui · Answer

Technically the first one has 5 answers and the second one has 10 answers since there are n-roots.

For instance, what's the square root of 4? There are two answers, -2 and +2. Similarly what are the fourth roots of 16? +2, -2, i2, and -i2. See? (i2)^4=16

The general formula for the n-roots of 1 is:$$e^{i 2\pi/n}=\cos(\frac{2\pi}{n})+i \sin(\frac{2\pi}{n})$$