Solve =2-i/5+7ika

$$= \frac { 2 - i } { 5 + 7 i } k a$$

Evaluate

$$\left(\frac{3}{74}-\frac{19}{74}i\right)ak$$

Solution Steps

Multiply both numerator and denominator of \frac{2-i}{5+7i} by the complex conjugate of the denominator, 5-7i.

$$\frac{\left(2-i\right)\left(5-7i\right)}{\left(5+7i\right)\left(5-7i\right)}ka$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2-i\right)\left(5-7i\right)}{5^{2}-7^{2}i^{2}}ka$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{\left(2-i\right)\left(5-7i\right)}{74}ka$$

Multiply complex numbers 2-i and 5-7i like you multiply binomials.

$$\frac{2\times 5+2\times \left(-7i\right)-i\times 5-\left(-7i^{2}\right)}{74}ka$$

By definition, i^{2} is -1.

$$\frac{2\times 5+2\times \left(-7i\right)-i\times 5-\left(-7\left(-1\right)\right)}{74}ka$$

Do the multiplications in 2\times 5+2\times \left(-7i\right)-i\times 5-\left(-7\left(-1\right)\right).

$$\frac{10-14i-5i-7}{74}ka$$

Combine the real and imaginary parts in 10-14i-5i-7.

$$\frac{10-7+\left(-14-5\right)i}{74}ka$$

Do the additions in 10-7+\left(-14-5\right)i.

$$\frac{3-19i}{74}ka$$

Divide 3-19i by 74 to get \frac{3}{74}-\frac{19}{74}i.

$$\left(\frac{3}{74}-\frac{19}{74}i\right)ka$$

Expand

$$\left(\frac{3}{74}-\frac{19}{74}i\right)ak$$

Solution Steps

Multiply both numerator and denominator of \frac{2-i}{5+7i} by the complex conjugate of the denominator, 5-7i.

$$\frac{\left(2-i\right)\left(5-7i\right)}{\left(5+7i\right)\left(5-7i\right)}ka$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2-i\right)\left(5-7i\right)}{5^{2}-7^{2}i^{2}}ka$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{\left(2-i\right)\left(5-7i\right)}{74}ka$$

Multiply complex numbers 2-i and 5-7i like you multiply binomials.

$$\frac{2\times 5+2\times \left(-7i\right)-i\times 5-\left(-7i^{2}\right)}{74}ka$$

By definition, i^{2} is -1.

$$\frac{2\times 5+2\times \left(-7i\right)-i\times 5-\left(-7\left(-1\right)\right)}{74}ka$$

Do the multiplications in 2\times 5+2\times \left(-7i\right)-i\times 5-\left(-7\left(-1\right)\right).

$$\frac{10-14i-5i-7}{74}ka$$

Combine the real and imaginary parts in 10-14i-5i-7.

$$\frac{10-7+\left(-14-5\right)i}{74}ka$$

Do the additions in 10-7+\left(-14-5\right)i.

$$\frac{3-19i}{74}ka$$

Divide 3-19i by 74 to get \frac{3}{74}-\frac{19}{74}i.

$$\left(\frac{3}{74}-\frac{19}{74}i\right)ka$$