Factor out <math><mstyle displaystyle="true"><msup><mrow><mi>i</mi></mrow><mrow><mn>20</mn></mrow></msup></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mi>i</mi></mrow><mrow><mn>20</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>i</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>5</mn></mrow></msup></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mi>i</mi></mrow><mrow><mn>4</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

One to any power is one.

Multiply <math><mstyle displaystyle="true"><msup><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

This is the trigonometric form of a complex number where <math><mstyle displaystyle="true"><mrow><mo>|</mo><mi>z</mi><mo>|</mo></mrow></mstyle></math> is the modulus and <math><mstyle displaystyle="true"><mi>θ</mi></mstyle></math> is the angle created on the complex plane.

The modulus of a complex number is the distance from the origin on the complex plane.

Substitute the actual values of <math><mstyle displaystyle="true"><mi>a</mi><mo>=</mo><mo>-</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mn>0</mn></mstyle></math> .

Raising <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to any positive power yields <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Any root of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

Since inverse tangent of <math><mstyle displaystyle="true"><mfrac><mrow><mn>0</mn></mrow><mrow><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></math> produces an angle in the second quadrant, the value of the angle is <math><mstyle displaystyle="true"><mi>π</mi></mstyle></math> .

Substitute the values of <math><mstyle displaystyle="true"><mi>θ</mi><mo>=</mo><mi>π</mi></mstyle></math> and <math><mstyle displaystyle="true"><mrow><mo>|</mo><mi>z</mi><mo>|</mo></mrow><mo>=</mo><mn>1</mn></mstyle></math> .

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