For the imaginary unit i, where i^2 = -1, which of the following is equivalent to the expr… — ACT

For the imaginary unit $i$ , where $i^2 = -1$ , which of the following is equivalent to the expression $\frac{5}{2 - i}$ ?

The simplified expression is

2 + i

To simplify the expression, multiply both the numerator and denominator by the complex conjugate of the denominator, which is

2 + i

. This results in

\frac{5(2+i)}{(2-i)(2+i)} = \frac{10+5i}{4-i^2}

. Since

i^2 = -1

, the denominator becomes

4 - (-1) = 5

. Dividing both terms in the numerator by

5

gives

2 + i

Multiply the numerator and the denominator by the complex conjugate of the denominator,

2 + i

\frac{5(2 + i)}{(2 - i)(2 + i)}

Multiplying by the conjugate rationalizes the denominator, converting it into a real number.

Expand the numerator and the denominator, substituting

-1

for

i^2

\frac{10 + 5i}{4 - (-1)} = \frac{10 + 5i}{5}

Using the distributive property for the numerator and the difference of squares identity for the denominator, along with the definition

i^2 = -1

Divide each term in the numerator by the denominator.

2 + i

Distributing the division by

5

to both the real and imaginary parts of the numerator simplifies the expression to standard form.

Rationalizing the denominator of a complex fraction by multiplying by the complex conjugate of the denominator.

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