Solve frac{i^{4}-2i^{2}+i^{6}-3i^9}{i^16-i^20+i^35}

$$\frac { i ^ { 4 } - 2 i ^ { 2 } + i ^ { 6 } - 3 i ^ { 9 } } { i ^ { 16 } - i ^ { 20 } + i ^ { 35 } }$$

Evaluate

$$3+2i$$

Solution Steps

Calculate i to the power of 4 and get 1.

$$\frac{1-2i^{2}+i^{6}-3i^{9}}{i^{16}-i^{20}+i^{35}}$$

Calculate i to the power of 2 and get -1.

$$\frac{1-2\left(-1\right)+i^{6}-3i^{9}}{i^{16}-i^{20}+i^{35}}$$

Multiply 2 and -1 to get -2.

$$\frac{1-\left(-2\right)+i^{6}-3i^{9}}{i^{16}-i^{20}+i^{35}}$$

The opposite of -2 is 2.

$$\frac{1+2+i^{6}-3i^{9}}{i^{16}-i^{20}+i^{35}}$$

Add 1 and 2 to get 3.

$$\frac{3+i^{6}-3i^{9}}{i^{16}-i^{20}+i^{35}}$$

Calculate i to the power of 6 and get -1.

$$\frac{3-1-3i^{9}}{i^{16}-i^{20}+i^{35}}$$

Subtract 1 from 3 to get 2.

$$\frac{2-3i^{9}}{i^{16}-i^{20}+i^{35}}$$

Calculate i to the power of 9 and get i.

$$\frac{2-3i}{i^{16}-i^{20}+i^{35}}$$

Calculate i to the power of 16 and get 1.

$$\frac{2-3i}{1-i^{20}+i^{35}}$$

Calculate i to the power of 20 and get 1.

$$\frac{2-3i}{1-1+i^{35}}$$

Subtract 1 from 1 to get 0.

$$\frac{2-3i}{0+i^{35}}$$

Calculate i to the power of 35 and get -i.

$$\frac{2-3i}{-i}$$

Multiply both numerator and denominator by imaginary unit i.

$$\frac{3+2i}{1}$$

Divide 3+2i by 1 to get 3+2i.

$$3+2i$$

Real Part

$$3$$

Solution Steps

Calculate i to the power of 4 and get 1.

$$Re(\frac{1-2i^{2}+i^{6}-3i^{9}}{i^{16}-i^{20}+i^{35}})$$

Calculate i to the power of 2 and get -1.

$$Re(\frac{1-2\left(-1\right)+i^{6}-3i^{9}}{i^{16}-i^{20}+i^{35}})$$

Multiply 2 and -1 to get -2.

$$Re(\frac{1-\left(-2\right)+i^{6}-3i^{9}}{i^{16}-i^{20}+i^{35}})$$

The opposite of -2 is 2.

$$Re(\frac{1+2+i^{6}-3i^{9}}{i^{16}-i^{20}+i^{35}})$$

Add 1 and 2 to get 3.

$$Re(\frac{3+i^{6}-3i^{9}}{i^{16}-i^{20}+i^{35}})$$

Calculate i to the power of 6 and get -1.

$$Re(\frac{3-1-3i^{9}}{i^{16}-i^{20}+i^{35}})$$

Subtract 1 from 3 to get 2.

$$Re(\frac{2-3i^{9}}{i^{16}-i^{20}+i^{35}})$$

Calculate i to the power of 9 and get i.

$$Re(\frac{2-3i}{i^{16}-i^{20}+i^{35}})$$

Calculate i to the power of 16 and get 1.

$$Re(\frac{2-3i}{1-i^{20}+i^{35}})$$

Calculate i to the power of 20 and get 1.

$$Re(\frac{2-3i}{1-1+i^{35}})$$

Subtract 1 from 1 to get 0.

$$Re(\frac{2-3i}{0+i^{35}})$$

Calculate i to the power of 35 and get -i.

$$Re(\frac{2-3i}{-i})$$

Multiply both numerator and denominator of \frac{2-3i}{-i} by imaginary unit i.

$$Re(\frac{3+2i}{1})$$

Divide 3+2i by 1 to get 3+2i.

$$Re(3+2i)$$

The real part of 3+2i is 3.

$$3$$