Adding exponents of negative one.

I've been working to find exact solutions to trig functions to brush up on my simplification skills. During my procedure, though, I came to a point in which I don't know how to advance. I have to simplify;
(-1)^(5/6) - (-1)^(1/6)
I know the answer, in advance, is -Sqrt(3), but, why? What procedure leads one to this answer? I looked up several tables of identities, but none had a case like this. Can anyone tell me what skill I'm missing?

Complex numbsers:
(-1)^(5/6) - (-1)^(1/6)
= -sqrt(3)/2 + i/2 - (sqrt(3)/2 + i/2)
= -2sqrt(3)/2
= -sqrt(3)

Edit:
(-1)^(5/6) - (-1)^(1/6)
= e^(5 i π/6) - e^(i π/6) => in polar form
Euler's formula:
e^(iθ) = cos(θ) + i sin(θ)
=> cos(5π/6) + i sin(5π/6) - cos(π/6) - i sin(π/6)
= -√3/2 + i * 1/2 - √3/2 - i * 1/2 => i/2 - i/2 => cancel out
= -√3/2 - √3/2 => add like terms
= -2√3/2 => 2's cancel
= -√3

You are welcome, and I'm glad it was helpful.

Regards.

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(-1)^(5/6) - (-1)^(1/6)

It will become like this :

(-1)^(5/6) + (1)^(-1/6)

[ ( -1 ) ( 1) ]^( 5/6)+(-1/6)

[ ( -1 ) ( 1) ]^( 5-1/6)

[ ( -1 ) ( 1) ]^( 4/6)

[ ( -1 ) ]^( 4/6)

( - 1 )^(4/6)

( -1 )^( 2/3 )

- ( 1 )^( 2/3)

Multiply 1 to the denominator of the exponent, which is 3
The answer, which is 3, add it to the numerator of the exponent, which is 2
The answer, which is 5, divide it to 3
The answer is 1.67, add -, as in -1.67 or -1.7 or -sqrt3

The -sqrt4 = -2
The -sqrt3 = -1.732050808 or -1.7 ( when rounded-off to one decimal place )
The -sqrt2 = -1.414213562 or - 1.4 ( when rounded-off to one decimal place )

( - 4 ) + ( - 2 ) = - 6
- 6 / 2 = - 3

( - 2 ) + ( - 1.414213562 ) = - 3.414213562
- 3.414213562 /2 = - 1.707106781 or - 1.7 ( when rounded-off to one decimal place )

In multiplication, add the exponents.
In division, subtract the exponents.

in the equation :
(-1)^(5/6) - (-1)^(1/6)
the sign outside the parentheses is negative which means subtraction.
In subtraction, change the sign of the subtrahend, from (-1)^(1/6) to (1)^(-1/6)
and proceed as in addition (-1)^(5/6) + (1)^(-1/6)
The starting amount is called minuend, the amount subtracted from is called the subtrahend.
Minuend (-1)^(5/6)
Subtrahend (-1)^(1/6)

(-1)^(5/6) - (-1)^(1/6)
= -Sqrt(3)

(-1)^(5/6) - (-1)^(1/6)

1. Subtract the exponents: (-1) ^ [ (5/6) - (1/6) ] = (-1) ^ (4/6)

2. Simplify: (-1) ^ (2/3)

Answer: -Sqrt(3)