Find the indicated power using De Moivre’s Theorem (−5−5sqrt 3i)^3

-5 - 5√3 i = -5 (1 + √3 i )

We have modulus

|-5 (1 + √3 i )| = 5 √(1² + (√3)²) = 5√4 = 10

and argument

arg(-5 - 5√3 i ) = π - arctan(√3) = 2π/3

(we subtract from π because the given complex number lies in the third quadrant of the complex plane, whereas the arctan function only returns angles between -π/2 and π/2)

so that the polar form of the number is

-5 - 5√3 i = 10 exp(2π/3 i )

By DeMoivre's theorem, we have

(-5 - 5√3 i )³ = 10³ exp(3 × 2π/3 i ) = 1000 exp(2πi ) = 1000

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step-by-step explanation:

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is