Solve z^3 = 27i - ScanMath

HOME CALCULATOR

DOWNLOAD FOR FREE

Algebra
Calculus
Trigonometry
Matrix

Type your math problem... z^3 = 27i

Question

Solve the equation

z_{1} = \frac{3 3}{2} + \frac{3}{2} i z_{2} = - \frac{3 3}{2} + \frac{3}{2} i z_{3} = - 3 i

Alternative Form

z_{1} \approx 2.598076 + 1.5 i z_{2} \approx - 2.598076 + 1.5 i z_{3} = - 3 i

Evaluate

z^{3} = 27 i

Simplify

z = 3 27 i

Rewrite the complex number in polar form

More Steps

z = 3 27 (cos (\frac{π}{2}) + i sin (\frac{π}{2}))

Calculate the nth roots of a complex r (cos (θ) + i \times sin (θ),using n z = n r (cos \frac{θ + 2 kπ}{n} + i sin \frac{θ + 2 kπ}{n})

z = 327 \times (cos (\frac{\frac{π}{2} + 2 kπ}{3}) + i sin (\frac{\frac{π}{2} + 2 kπ}{3}))

Simplify

z = 3 (cos (\frac{\frac{π}{2} + 2 kπ}{3}) + i sin (\frac{\frac{π}{2} + 2 kπ}{3}))

Since n = 3,substitute k = 0, 1, 2 into the expression

z_{1} = 3 (cos (\frac{\frac{π}{2} + 2 \times 0 \times π}{3}) + i sin (\frac{\frac{π}{2} + 2 \times 0 \times π}{3})) z_{2} = 3 (cos (\frac{\frac{π}{2} + 2 \times 1 \times π}{3}) + i sin (\frac{\frac{π}{2} + 2 \times 1 \times π}{3})) z_{3} = 3 (cos (\frac{\frac{π}{2} + 2 \times 2 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \times 2 π}{3}))

Calculate

More Steps

z_{1} = 3 (cos (\frac{π}{6}) + i sin (\frac{π}{6})) z_{2} = 3 (cos (\frac{\frac{π}{2} + 2 \times 1 \times π}{3}) + i sin (\frac{\frac{π}{2} + 2 \times 1 \times π}{3})) z_{3} = 3 (cos (\frac{\frac{π}{2} + 2 \times 2 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \times 2 π}{3}))

Calculate

More Steps

z_{1} = 3 (cos (\frac{π}{6}) + i sin (\frac{π}{6})) z_{2} = 3 (cos (\frac{5 π}{6}) + i sin (\frac{5 π}{6})) z_{3} = 3 (cos (\frac{\frac{π}{2} + 2 \times 2 π}{3}) + i sin (\frac{\frac{π}{2} + 2 \times 2 π}{3}))

Calculate

More Steps

z_{1} = 3 (cos (\frac{π}{6}) + i sin (\frac{π}{6})) z_{2} = 3 (cos (\frac{5 π}{6}) + i sin (\frac{5 π}{6})) z_{3} = 3 (cos (\frac{3 π}{2}) + i sin (\frac{3 π}{2}))

Solution

z_{1} = \frac{3 3}{2} + \frac{3}{2} i z_{2} = - \frac{3 3}{2} + \frac{3}{2} i z_{3} = - 3 i

Alternative Form

z_{1} \approx 2.598076 + 1.5 i z_{2} \approx - 2.598076 + 1.5 i z_{3} = - 3 i

Show Solution