Solve (3i^4-2i^35)+(1+5i^14)-(9-23i^7)

$$( 3 i ^ { 4 } - 2 i ^ { 35 } ) + ( 1 + 5 i ^ { 14 } ) - ( 9 - 23 i ^ { 7 } )$$

Evaluate

$$-10-21i$$

Solution Steps

Calculate i to the power of 4 and get 1.

$$3\times 1-2i^{35}+1+5i^{14}-\left(9-23i^{7}\right)$$

Multiply 3 and 1 to get 3.

$$3-2i^{35}+1+5i^{14}-\left(9-23i^{7}\right)$$

Calculate i to the power of 35 and get -i.

$$3-2\left(-i\right)+1+5i^{14}-\left(9-23i^{7}\right)$$

Multiply 2 and -i to get -2i.

$$3-\left(-2i\right)+1+5i^{14}-\left(9-23i^{7}\right)$$

The opposite of -2i is 2i.

$$3+2i+1+5i^{14}-\left(9-23i^{7}\right)$$

Do the additions in 3+2i+1.

$$5i^{14}+4+2i-\left(9-23i^{7}\right)$$

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

$$5i^{14}+4+2i-\left(9-23\left(-i\right)\right)$$

Multiply 23 and -i to get -23i.

$$5i^{14}+4+2i-\left(9-\left(-23i\right)\right)$$

The opposite of -23i is 23i.

$$5i^{14}+4+2i-\left(9+23i\right)$$

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

$$5\left(-1\right)+4+2i-\left(9+23i\right)$$

Multiply 5 and -1 to get -5.

$$-5+4+2i-\left(9+23i\right)$$

Do the additions in -5+4+2i.

$$-1+2i-\left(9+23i\right)$$

Subtract 9+23i from -1+2i to get -10-21i.

$$-10-21i$$

Real Part

$$-10$$

Solution Steps

Calculate i to the power of 4 and get 1.

$$Re(3\times 1-2i^{35}+1+5i^{14}-\left(9-23i^{7}\right))$$

Multiply 3 and 1 to get 3.

$$Re(3-2i^{35}+1+5i^{14}-\left(9-23i^{7}\right))$$

Calculate i to the power of 35 and get -i.

$$Re(3-2\left(-i\right)+1+5i^{14}-\left(9-23i^{7}\right))$$

Multiply 2 and -i to get -2i.

$$Re(3-\left(-2i\right)+1+5i^{14}-\left(9-23i^{7}\right))$$

The opposite of -2i is 2i.

$$Re(3+2i+1+5i^{14}-\left(9-23i^{7}\right))$$

Do the additions in 3+2i+1.

$$Re(5i^{14}+4+2i-\left(9-23i^{7}\right))$$

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

$$Re(5i^{14}+4+2i-\left(9-23\left(-i\right)\right))$$

Multiply 23 and -i to get -23i.

$$Re(5i^{14}+4+2i-\left(9-\left(-23i\right)\right))$$

The opposite of -23i is 23i.

$$Re(5i^{14}+4+2i-\left(9+23i\right))$$

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

$$Re(5\left(-1\right)+4+2i-\left(9+23i\right))$$

Multiply 5 and -1 to get -5.

$$Re(-5+4+2i-\left(9+23i\right))$$

Do the additions in -5+4+2i.

$$Re(-1+2i-\left(9+23i\right))$$

Subtract 9+23i from -1+2i to get -10-21i.

$$Re(-10-21i)$$

The real part of -10-21i is -10.

$$-10$$