Solve [2i-3(-i)^{3}]^{5}+[(-2i)^3]^5+(3i^7-5i)^5

$$[ 2 i - 3 ( - i ) ^ { 3 } ] ^ { 5 } + [ ( - 2 i ) ^ { 3 } ] ^ { 5 } + ( 3 i ^ { 7 } - 5 i ) ^ { 5 }$$

Evaluate

$$-i$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and 5 to get 15.

$$\left(2i-3\left(-i\right)^{3}\right)^{5}+\left(-2i\right)^{15}+\left(3i^{7}-5i\right)^{5}$$

Calculate -i to the power of 3 and get i.

$$\left(2i-3i\right)^{5}+\left(-2i\right)^{15}+\left(3i^{7}-5i\right)^{5}$$

Subtract 3i from 2i to get -i.

$$\left(-i\right)^{5}+\left(-2i\right)^{15}+\left(3i^{7}-5i\right)^{5}$$

Calculate -i to the power of 5 and get -i.

$$-i+\left(-2i\right)^{15}+\left(3i^{7}-5i\right)^{5}$$

Calculate -2i to the power of 15 and get 32768i.

$$-i+32768i+\left(3i^{7}-5i\right)^{5}$$

Add -i and 32768i to get 32767i.

$$32767i+\left(3i^{7}-5i\right)^{5}$$

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

$$32767i+\left(3\left(-i\right)-5i\right)^{5}$$

Multiply 3 and -i to get -3i.

$$32767i+\left(-3i-5i\right)^{5}$$

Subtract 5i from -3i to get -8i.

$$32767i+\left(-8i\right)^{5}$$

Calculate -8i to the power of 5 and get -32768i.

$$32767i-32768i$$

Subtract 32768i from 32767i to get -i.

$$-i$$

Real Part

$$0$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and 5 to get 15.

$$Re(\left(2i-3\left(-i\right)^{3}\right)^{5}+\left(-2i\right)^{15}+\left(3i^{7}-5i\right)^{5})$$

Calculate -i to the power of 3 and get i.

$$Re(\left(2i-3i\right)^{5}+\left(-2i\right)^{15}+\left(3i^{7}-5i\right)^{5})$$

Subtract 3i from 2i to get -i.

$$Re(\left(-i\right)^{5}+\left(-2i\right)^{15}+\left(3i^{7}-5i\right)^{5})$$

Calculate -i to the power of 5 and get -i.

$$Re(-i+\left(-2i\right)^{15}+\left(3i^{7}-5i\right)^{5})$$

Calculate -2i to the power of 15 and get 32768i.

$$Re(-i+32768i+\left(3i^{7}-5i\right)^{5})$$

Add -i and 32768i to get 32767i.

$$Re(32767i+\left(3i^{7}-5i\right)^{5})$$

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

$$Re(32767i+\left(3\left(-i\right)-5i\right)^{5})$$

Multiply 3 and -i to get -3i.

$$Re(32767i+\left(-3i-5i\right)^{5})$$

Subtract 5i from -3i to get -8i.

$$Re(32767i+\left(-8i\right)^{5})$$

Calculate -8i to the power of 5 and get -32768i.

$$Re(32767i-32768i)$$

Subtract 32768i from 32767i to get -i.

$$Re(-i)$$

The real part of -i is 0.

$$0$$