Solve (13i-4)m^5-7i=i - ScanMath

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Question

Solve the equation

m_{1} = 58 \times cos (\frac{π}{10}) + 58 \times sin (\frac{π}{10}) \times i m_{2} = 58 \times i m_{3} = - 58 \times cos (\frac{π}{10}) + 58 \times sin (\frac{9 π}{10}) \times i m_{4} = 58 \times cos (\frac{13 π}{10}) + 58 \times sin (\frac{13 π}{10}) \times i m_{5} = 58 \times cos (\frac{17 π}{10}) + 58 \times sin (\frac{17 π}{10}) \times i

Alternative Form

m_{1} = 58 \times cos (1 8^{\circ}) + 58 \times sin (1 8^{\circ}) \times i m_{2} = 58 \times i m_{3} = - 58 \times cos (1 8^{\circ}) + 58 \times sin (16 2^{\circ}) \times i m_{4} = 58 \times cos (23 4^{\circ}) + 58 \times sin (23 4^{\circ}) \times i m_{5} = 58 \times cos (30 6^{\circ}) + 58 \times sin (30 6^{\circ}) \times i

Alternative Form

m_{1} \approx 1.441532 + 0.468382 i m_{2} \approx 1.515717 i m_{3} \approx - 1.441532 + 0.468382 i m_{4} \approx - 0.890916 - 1.22624 i m_{5} \approx 0.890916 - 1.22624 i

Evaluate

(13 i - 4) m^{5} - 7 i = i

Calculate

(- 4 + 13 i) m^{5} - 7 i = i

Move the constant to the right-hand side and change its sign

(- 4 + 13 i) m^{5} = i + 7 i

Add the numbers

More Steps

(- 4 + 13 i) m^{5} = 8 i

Simplify

m = 5 8 i

Rewrite the complex number in polar form

More Steps

m = 5 8 (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})

m = 58 \times (cos (\frac{\frac{π}{2} + 2 kπ}{5}) + i sin (\frac{\frac{π}{2} + 2 kπ}{5}))

Since n = 5,substitute k = 0, 1, 2, 3, 4 into the expression

m_{1} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 0 \times π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 0 \times π}{5})) m_{2} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 1 \times π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 1 \times π}{5})) m_{3} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 2 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 2 π}{5})) m_{4} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 3 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 3 π}{5})) m_{5} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 4 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 4 π}{5}))

Calculate

More Steps

m_{1} = 58 \times (cos (\frac{π}{10}) + i sin (\frac{π}{10})) m_{2} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 1 \times π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 1 \times π}{5})) m_{3} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 2 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 2 π}{5})) m_{4} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 3 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 3 π}{5})) m_{5} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 4 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 4 π}{5}))

Calculate

More Steps

m_{1} = 58 \times (cos (\frac{π}{10}) + i sin (\frac{π}{10})) m_{2} = 58 \times (cos (\frac{π}{2}) + i sin (\frac{π}{2})) m_{3} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 2 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 2 π}{5})) m_{4} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 3 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 3 π}{5})) m_{5} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 4 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 4 π}{5}))

Calculate

More Steps

m_{1} = 58 \times (cos (\frac{π}{10}) + i sin (\frac{π}{10})) m_{2} = 58 \times (cos (\frac{π}{2}) + i sin (\frac{π}{2})) m_{3} = 58 \times (cos (\frac{9 π}{10}) + i sin (\frac{9 π}{10})) m_{4} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 3 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 3 π}{5})) m_{5} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 4 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 4 π}{5}))

Calculate

More Steps

m_{1} = 58 \times (cos (\frac{π}{10}) + i sin (\frac{π}{10})) m_{2} = 58 \times (cos (\frac{π}{2}) + i sin (\frac{π}{2})) m_{3} = 58 \times (cos (\frac{9 π}{10}) + i sin (\frac{9 π}{10})) m_{4} = 58 \times (cos (\frac{13 π}{10}) + i sin (\frac{13 π}{10})) m_{5} = 58 \times (cos (\frac{\frac{π}{2} + 2 \times 4 π}{5}) + i sin (\frac{\frac{π}{2} + 2 \times 4 π}{5}))

Calculate

More Steps

m_{1} = 58 \times (cos (\frac{π}{10}) + i sin (\frac{π}{10})) m_{2} = 58 \times (cos (\frac{π}{2}) + i sin (\frac{π}{2})) m_{3} = 58 \times (cos (\frac{9 π}{10}) + i sin (\frac{9 π}{10})) m_{4} = 58 \times (cos (\frac{13 π}{10}) + i sin (\frac{13 π}{10})) m_{5} = 58 \times (cos (\frac{17 π}{10}) + i sin (\frac{17 π}{10}))

Solution

m_{1} = 58 \times cos (\frac{π}{10}) + 58 \times sin (\frac{π}{10}) \times i m_{2} = 58 \times i m_{3} = - 58 \times cos (\frac{π}{10}) + 58 \times sin (\frac{9 π}{10}) \times i m_{4} = 58 \times cos (\frac{13 π}{10}) + 58 \times sin (\frac{13 π}{10}) \times i m_{5} = 58 \times cos (\frac{17 π}{10}) + 58 \times sin (\frac{17 π}{10}) \times i

Alternative Form

m_{1} = 58 \times cos (1 8^{\circ}) + 58 \times sin (1 8^{\circ}) \times i m_{2} = 58 \times i m_{3} = - 58 \times cos (1 8^{\circ}) + 58 \times sin (16 2^{\circ}) \times i m_{4} = 58 \times cos (23 4^{\circ}) + 58 \times sin (23 4^{\circ}) \times i m_{5} = 58 \times cos (30 6^{\circ}) + 58 \times sin (30 6^{\circ}) \times i

Alternative Form

m_{1} \approx 1.441532 + 0.468382 i m_{2} \approx 1.515717 i m_{3} \approx - 1.441532 + 0.468382 i m_{4} \approx - 0.890916 - 1.22624 i m_{5} \approx 0.890916 - 1.22624 i

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