Question

simplify.
\\(\frac{1}{5 - 2i}\\)
write your answer in the form \\(a + bi\\). reduce all fractions.

Explanation:

Step1: Multiply by conjugate

Multiply numerator and denominator by the conjugate of the denominator, $5+2i$:
$$\frac{1}{5-2i} \times \frac{5+2i}{5+2i} = \frac{5+2i}{(5-2i)(5+2i)}$$

Step2: Expand denominator

Use difference of squares $(a-b)(a+b)=a^2-b^2$, with $i^2=-1$:
$$(5-2i)(5+2i) = 5^2 - (2i)^2 = 25 - 4i^2 = 25 - 4(-1) = 25 + 4 = 29$$

Step3: Split the fraction

Separate the real and imaginary parts:
$$\frac{5+2i}{29} = \frac{5}{29} + \frac{2}{29}i$$

Answer:

$\frac{5}{29} + \frac{2}{29}i$