To find the given power of (-1/2 - sqrt(3)/2i)^10 in rectangular form, we can use De Moivre's formula. The expression can be simplified to cos(10π/3) - i sin(10π/3).
To find the given power of (-1/2 - sqrt(3)/2i)^10 in rectangular form, we need to understand two concepts: exponentiation and complex numbers.
First, we need to raise (-1/2 - sqrt(3)/2i) to the tenth power.
To do this, we can use the binomial theorem or the De Moivre's formula.
The binomial theorem expands the expression using binomial coefficients, while De Moivre's formula gives a succinct way to raise complex numbers to powers.
Using De Moivre's formula, we can write (-1/2 - sqrt(3)/2i)^10 as:
[ cos(θ) + i sin(θ) ]^10
where θ = -π/3.
Expanding the formula using the binomial theorem, we have:
[ cos(θ) + i sin(θ) ]^10 = cos(10θ) + i sin(10θ)
Substituting θ = -π/3, we get:
cos(10(-π/3)) + i sin(10(-π/3))
Simplifying the expression further, we have:
cos(-10π/3) + i sin(-10π/3)
Using the trigonometric identity cos(-x) = cos(x) and sin(-x) = -sin(x), we can simplify it as:
cos(10π/3) - i sin(10π/3)
Therefore, the given power (-1/2 - sqrt(3)/2i)^10, in rectangular form, is:
cos(10π/3) - i sin(10π/3).