Solve 2-2i+5i^5+6i^6+7i^7+8i^8

$$2 - 2 i + 5 i ^ { 5 } + 6 i ^ { 6 } + 7 i ^ { 7 } + 8 i ^ { 8 }$$

Evaluate

$$4-4i$$

Solution Steps

Calculate i to the power of 5 and get i.

$$2-2i+5i+6i^{6}+7i^{7}+8i^{8}$$

Do the additions.

$$6i^{6}+7i^{7}+8i^{8}+2+3i$$

Calculate i to the power of 8 and get 1.

$$6i^{6}+7i^{7}+8\times 1+2+3i$$

Multiply 8 and 1 to get 8.

$$6i^{6}+7i^{7}+8+2+3i$$

Do the additions.

$$6i^{6}+7i^{7}+10+3i$$

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

$$6\left(-1\right)+7i^{7}+10+3i$$

Multiply 6 and -1 to get -6.

$$-6+7i^{7}+10+3i$$

Do the additions.

$$7i^{7}+4+3i$$

Calculate i to the power of 7 and get -i.

$$7\left(-i\right)+4+3i$$

Multiply 7 and -i to get -7i.

$$-7i+4+3i$$

Do the additions.

$$4-4i$$

Real Part

$$4$$

Solution Steps

Calculate i to the power of 5 and get i.

$$Re(2-2i+5i+6i^{6}+7i^{7}+8i^{8})$$

Do the additions in 2-2i+5i.

$$Re(6i^{6}+7i^{7}+8i^{8}+2+3i)$$

Calculate i to the power of 8 and get 1.

$$Re(6i^{6}+7i^{7}+8\times 1+2+3i)$$

Multiply 8 and 1 to get 8.

$$Re(6i^{6}+7i^{7}+8+2+3i)$$

Do the additions in 8+2+3i.

$$Re(6i^{6}+7i^{7}+10+3i)$$

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

$$Re(6\left(-1\right)+7i^{7}+10+3i)$$

Multiply 6 and -1 to get -6.

$$Re(-6+7i^{7}+10+3i)$$

Do the additions in -6+10+3i.

$$Re(7i^{7}+4+3i)$$

Calculate i to the power of 7 and get -i.

$$Re(7\left(-i\right)+4+3i)$$

Multiply 7 and -i to get -7i.

$$Re(-7i+4+3i)$$

Do the additions in -7i+4+3i.

$$Re(4-4i)$$

The real part of 4-4i is 4.

$$4$$