Solve 1/(2+i)^7

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

Evaluate

$$-\frac{278}{78125}+\frac{29}{78125}i=-0.0035584+0.0003712i$$

Steps for Raising Power to Another Power

Use the rules of exponents to simplify the expression.

$$\frac{1}{\left(2+i\right)^{7}}$$

To raise a power to another power, multiply the exponents.

$$\left(2+i\right)^{7\left(-1\right)}$$

Multiply 7 times -1.

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

Raise 2+i to the power -7.

$$-\frac{278}{78125}+\frac{29}{78125}i$$

Real Part

$$-\frac{278}{78125} = -0.0035584$$

Solution Steps

Calculate 2+i to the power of 7 and get -278-29i.

$$Re(\frac{1}{-278-29i})$$

Multiply both numerator and denominator of \frac{1}{-278-29i} by the complex conjugate of the denominator, -278+29i.

$$Re(\frac{1\left(-278+29i\right)}{\left(-278-29i\right)\left(-278+29i\right)})$$

Do the multiplications in \frac{1\left(-278+29i\right)}{\left(-278-29i\right)\left(-278+29i\right)}.

$$Re(\frac{-278+29i}{78125})$$

Divide -278+29i by 78125 to get -\frac{278}{78125}+\frac{29}{78125}i.

$$Re(-\frac{278}{78125}+\frac{29}{78125}i)$$

The real part of -\frac{278}{78125}+\frac{29}{78125}i is -\frac{278}{78125}.

$$-\frac{278}{78125}$$