Solve (4+3i/3-4i)^10

$$( \frac { 4 + 3 i } { 3 - 4 i } ) ^ { 10 }$$

Evaluate

$$-1$$

Solution Steps

Multiply both numerator and denominator of \frac{4+3i}{3-4i} by the complex conjugate of the denominator, 3+4i.

$$\left(\frac{\left(4+3i\right)\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}\right)^{10}$$

Do the multiplications in \frac{\left(4+3i\right)\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}.

$$\left(\frac{25i}{25}\right)^{10}$$

Divide 25i by 25 to get i.

$$i^{10}$$

Calculate i to the power of 10 and get -1.

$$-1$$

Real Part

$$-1$$

Solution Steps

Multiply both numerator and denominator of \frac{4+3i}{3-4i} by the complex conjugate of the denominator, 3+4i.

$$Re(\left(\frac{\left(4+3i\right)\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}\right)^{10})$$

Do the multiplications in \frac{\left(4+3i\right)\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}.

$$Re(\left(\frac{25i}{25}\right)^{10})$$

Divide 25i by 25 to get i.

$$Re(i^{10})$$

Calculate i to the power of 10 and get -1.

$$Re(-1)$$

The real part of -1 is -1.

$$-1$$