College

Write the expression in the standard form [tex]a + bi[/tex].

[tex]\frac{4}{4+3i}[/tex]

A. [tex]\frac{16}{7} - \frac{12}{7}i[/tex]

B. [tex]\frac{16}{25} + \frac{12}{25}i[/tex]

C. [tex]\frac{16}{25} - \frac{12}{25}i[/tex]

D. [tex]\frac{16}{7} + \frac{12}{7}i[/tex]

Answer :

To write the complex number in the form [tex]\(a + bi\)[/tex], we start with

[tex]$$
\frac{4}{4+3i}.
$$[/tex]

Step 1. Multiply by the conjugate

Multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(4-3i\)[/tex]:

[tex]$$
\frac{4}{4+3i} \cdot \frac{4-3i}{4-3i}.
$$[/tex]

Step 2. Simplify the numerator

Multiply the numerator:

[tex]$$
4(4-3i) = 16 - 12i.
$$[/tex]

Step 3. Simplify the denominator

Multiply the denominator using the difference of squares formula:

[tex]$$
(4+3i)(4-3i) = 4^2 - (3i)^2.
$$[/tex]

Remember that [tex]\(i^2 = -1\)[/tex], so

[tex]$$
(3i)^2 = 9i^2 = 9(-1) = -9.
$$[/tex]

Thus,

[tex]$$
4^2 - (3i)^2 = 16 - (-9) = 16 + 9 = 25.
$$[/tex]

Step 4. Write in standard form

Now, the expression becomes:

[tex]$$
\frac{16 - 12i}{25} = \frac{16}{25} - \frac{12}{25}i.
$$[/tex]

This is the standard form [tex]\(a + bi\)[/tex] where

[tex]$$
a = \frac{16}{25} \quad \text{and} \quad b = -\frac{12}{25}.
$$[/tex]

Conclusion

The expression written in the form [tex]\(a + bi\)[/tex] is:

[tex]$$
\frac{16}{25} - \frac{12}{25} i,
$$[/tex]

which corresponds to option C.