Solve frac{i^{4}+i^{9}+i^16}{2-i^5+i^10-i^15}

$$\frac { i ^ { 4 } + i ^ { 9 } + i ^ { 16 } } { 2 - i ^ { 5 } + i ^ { 10 } - i ^ { 15 } }$$

Evaluate

$$2+i$$

Solution Steps

Calculate i to the power of 4 and get 1.

$$\frac{1+i^{9}+i^{16}}{2-i^{5}+i^{10}-i^{15}}$$

Calculate i to the power of 9 and get i.

$$\frac{1+i+i^{16}}{2-i^{5}+i^{10}-i^{15}}$$

Calculate i to the power of 16 and get 1.

$$\frac{1+i+1}{2-i^{5}+i^{10}-i^{15}}$$

Do the additions in 1+i+1.

$$\frac{2+i}{2-i^{5}+i^{10}-i^{15}}$$

Calculate i to the power of 5 and get i.

$$\frac{2+i}{2-i+i^{10}-i^{15}}$$

Calculate i to the power of 10 and get -1.

$$\frac{2+i}{2-i-1-i^{15}}$$

Subtract 1 from 2-i to get 1-i.

$$\frac{2+i}{1-i-i^{15}}$$

Calculate i to the power of 15 and get -i.

$$\frac{2+i}{1-i-\left(-i\right)}$$

The opposite of -i is i.

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

Add 1-i and i to get 1.

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

Anything divided by one gives itself.

$$2+i$$

Real Part

$$2$$

Solution Steps

Calculate i to the power of 4 and get 1.

$$Re(\frac{1+i^{9}+i^{16}}{2-i^{5}+i^{10}-i^{15}})$$

Calculate i to the power of 9 and get i.

$$Re(\frac{1+i+i^{16}}{2-i^{5}+i^{10}-i^{15}})$$

Calculate i to the power of 16 and get 1.

$$Re(\frac{1+i+1}{2-i^{5}+i^{10}-i^{15}})$$

Do the additions in 1+i+1.

$$Re(\frac{2+i}{2-i^{5}+i^{10}-i^{15}})$$

Calculate i to the power of 5 and get i.

$$Re(\frac{2+i}{2-i+i^{10}-i^{15}})$$

Calculate i to the power of 10 and get -1.

$$Re(\frac{2+i}{2-i-1-i^{15}})$$

Subtract 1 from 2-i to get 1-i.

$$Re(\frac{2+i}{1-i-i^{15}})$$

Calculate i to the power of 15 and get -i.

$$Re(\frac{2+i}{1-i-\left(-i\right)})$$

The opposite of -i is i.

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

Add 1-i and i to get 1.

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

Anything divided by one gives itself.

$$Re(2+i)$$

The real part of 2+i is 2.

$$2$$