Solve frac{7i^{9}+2i^{8}-i^7}{i^6-i^5+i^4}

$$\frac { 7 i ^ { 9 } + 2 i ^ { 8 } - i ^ { 7 } } { i ^ { 6 } - i ^ { 5 } + i ^ { 4 } }$$

Evaluate

$$-8+2i$$

Solution Steps

Calculate i to the power of 9 and get i.

$$\frac{7i+2i^{8}-i^{7}}{i^{6}-i^{5}+i^{4}}$$

Calculate i to the power of 8 and get 1.

$$\frac{7i+2\times 1-i^{7}}{i^{6}-i^{5}+i^{4}}$$

Multiply 2 and 1 to get 2.

$$\frac{7i+2-i^{7}}{i^{6}-i^{5}+i^{4}}$$

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

$$\frac{7i+2-\left(-i\right)}{i^{6}-i^{5}+i^{4}}$$

The opposite of -i is i.

$$\frac{7i+2+i}{i^{6}-i^{5}+i^{4}}$$

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

$$\frac{7i+2+i}{-1-i^{5}+i^{4}}$$

Calculate i to the power of 5 and get i.

$$\frac{7i+2+i}{-1-i+i^{4}}$$

Calculate i to the power of 4 and get 1.

$$\frac{7i+2+i}{-1-i+1}$$

Add -1-i and 1 to get -i.

$$\frac{7i+2+i}{-i}$$

Do the additions in 7i+2+i.

$$\frac{2+8i}{-i}$$

Multiply both numerator and denominator by imaginary unit i.

$$\frac{-8+2i}{1}$$

Divide -8+2i by 1 to get -8+2i.

$$-8+2i$$

Real Part

$$-8$$

Solution Steps

Calculate i to the power of 9 and get i.

$$Re(\frac{7i+2i^{8}-i^{7}}{i^{6}-i^{5}+i^{4}})$$

Calculate i to the power of 8 and get 1.

$$Re(\frac{7i+2\times 1-i^{7}}{i^{6}-i^{5}+i^{4}})$$

Multiply 2 and 1 to get 2.

$$Re(\frac{7i+2-i^{7}}{i^{6}-i^{5}+i^{4}})$$

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

$$Re(\frac{7i+2-\left(-i\right)}{i^{6}-i^{5}+i^{4}})$$

The opposite of -i is i.

$$Re(\frac{7i+2+i}{i^{6}-i^{5}+i^{4}})$$

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

$$Re(\frac{7i+2+i}{-1-i^{5}+i^{4}})$$

Calculate i to the power of 5 and get i.

$$Re(\frac{7i+2+i}{-1-i+i^{4}})$$

Calculate i to the power of 4 and get 1.

$$Re(\frac{7i+2+i}{-1-i+1})$$

Add -1-i and 1 to get -i.

$$Re(\frac{7i+2+i}{-i})$$

Do the additions in 7i+2+i.

$$Re(\frac{2+8i}{-i})$$

Multiply both numerator and denominator of \frac{2+8i}{-i} by imaginary unit i.

$$Re(\frac{-8+2i}{1})$$

Divide -8+2i by 1 to get -8+2i.

$$Re(-8+2i)$$

The real part of -8+2i is -8.

$$-8$$