Solve frac{1^{80}+i^{12}-3i^26+2i^14}{9+2i-1^35*1^9}

$$\frac { 1 ^ { 80 } + i ^ { 12 } - 3 i ^ { 26 } + 2 i ^ { 14 } } { 9 + 2 i - 1 ^ { 35 } \cdot 1 ^ { 9 } }$$

Evaluate

$$\frac{6}{17}-\frac{3}{34}i\approx 0.352941176-0.088235294i$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 35 and 9 to get 44.

$$\frac{1^{80}+i^{12}-3i^{26}+2i^{14}}{9+2i-1^{44}}$$

Calculate 1 to the power of 80 and get 1.

$$\frac{1+i^{12}-3i^{26}+2i^{14}}{9+2i-1^{44}}$$

Calculate i to the power of 12 and get 1.

$$\frac{1+1-3i^{26}+2i^{14}}{9+2i-1^{44}}$$

Add 1 and 1 to get 2.

$$\frac{2-3i^{26}+2i^{14}}{9+2i-1^{44}}$$

Calculate i to the power of 26 and get -1.

$$\frac{2-3\left(-1\right)+2i^{14}}{9+2i-1^{44}}$$

Multiply 3 and -1 to get -3.

$$\frac{2-\left(-3\right)+2i^{14}}{9+2i-1^{44}}$$

The opposite of -3 is 3.

$$\frac{2+3+2i^{14}}{9+2i-1^{44}}$$

Add 2 and 3 to get 5.

$$\frac{5+2i^{14}}{9+2i-1^{44}}$$

Calculate i to the power of 14 and get -1.

$$\frac{5+2\left(-1\right)}{9+2i-1^{44}}$$

Multiply 2 and -1 to get -2.

$$\frac{5-2}{9+2i-1^{44}}$$

Subtract 2 from 5 to get 3.

$$\frac{3}{9+2i-1^{44}}$$

Calculate 1 to the power of 44 and get 1.

$$\frac{3}{9+2i-1}$$

Subtract 1 from 9+2i to get 8+2i.

$$\frac{3}{8+2i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 8-2i.

$$\frac{3\left(8-2i\right)}{\left(8+2i\right)\left(8-2i\right)}$$

Do the multiplications in \frac{3\left(8-2i\right)}{\left(8+2i\right)\left(8-2i\right)}.

$$\frac{24-6i}{68}$$

Divide 24-6i by 68 to get \frac{6}{17}-\frac{3}{34}i.

$$\frac{6}{17}-\frac{3}{34}i$$

Real Part

$$\frac{6}{17} = 0.35294117647058826$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 35 and 9 to get 44.

$$Re(\frac{1^{80}+i^{12}-3i^{26}+2i^{14}}{9+2i-1^{44}})$$

Calculate 1 to the power of 80 and get 1.

$$Re(\frac{1+i^{12}-3i^{26}+2i^{14}}{9+2i-1^{44}})$$

Calculate i to the power of 12 and get 1.

$$Re(\frac{1+1-3i^{26}+2i^{14}}{9+2i-1^{44}})$$

Add 1 and 1 to get 2.

$$Re(\frac{2-3i^{26}+2i^{14}}{9+2i-1^{44}})$$

Calculate i to the power of 26 and get -1.

$$Re(\frac{2-3\left(-1\right)+2i^{14}}{9+2i-1^{44}})$$

Multiply 3 and -1 to get -3.

$$Re(\frac{2-\left(-3\right)+2i^{14}}{9+2i-1^{44}})$$

The opposite of -3 is 3.

$$Re(\frac{2+3+2i^{14}}{9+2i-1^{44}})$$

Add 2 and 3 to get 5.

$$Re(\frac{5+2i^{14}}{9+2i-1^{44}})$$

Calculate i to the power of 14 and get -1.

$$Re(\frac{5+2\left(-1\right)}{9+2i-1^{44}})$$

Multiply 2 and -1 to get -2.

$$Re(\frac{5-2}{9+2i-1^{44}})$$

Subtract 2 from 5 to get 3.

$$Re(\frac{3}{9+2i-1^{44}})$$

Calculate 1 to the power of 44 and get 1.

$$Re(\frac{3}{9+2i-1})$$

Subtract 1 from 9+2i to get 8+2i.

$$Re(\frac{3}{8+2i})$$

Multiply both numerator and denominator of \frac{3}{8+2i} by the complex conjugate of the denominator, 8-2i.

$$Re(\frac{3\left(8-2i\right)}{\left(8+2i\right)\left(8-2i\right)})$$

Do the multiplications in \frac{3\left(8-2i\right)}{\left(8+2i\right)\left(8-2i\right)}.

$$Re(\frac{24-6i}{68})$$

Divide 24-6i by 68 to get \frac{6}{17}-\frac{3}{34}i.

$$Re(\frac{6}{17}-\frac{3}{34}i)$$

The real part of \frac{6}{17}-\frac{3}{34}i is \frac{6}{17}.

$$\frac{6}{17}$$