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Express as a complex number in simplest a+bi form:

(4-3i)/(-4-4i)
Answer:

Express as a complex number in simplest a+bi form: $\newline$ $\frac{4-3 i}{-4-4 i}$ $\newline$ Answer:

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Even and odd functions

Full solution

Q. Express as a complex number in simplest a+bi form:

\newline

\frac{4-3 i}{-4-4 i}

\newline

Answer:

Identify Conjugate: To simplify a complex fraction, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a + bi$ is $a - bi$ . So, the conjugate of $-4 - 4i$ is $-4 + 4i$ .
Multiply Numerator and Denominator: Now, we multiply both the numerator and the denominator by the conjugate of the denominator: $\newline$ $(4 - 3i) \times (-4 + 4i) / (-4 - 4i) \times (-4 + 4i)$
Perform Multiplication in Numerator: We perform the multiplication in the numerator: $\newline$ $(4 \times -4) + (4 \times 4i) + (-3i \times -4) + (-3i \times 4i) = -16 + 16i + 12i - 12i^2$ $\newline$ Since $i^2 = -1$ , we replace $-12i^2$ with $12$ : $\newline$ $-16 + 16i + 12i + 12 = -4 + 28i$
Perform Multiplication in Denominator: We perform the multiplication in the denominator: $\newline$ $(-4 \times -4) + (-4 \times 4i) + (-4i \times -4) + (-4i \times 4i) = 16 - 16i + 16i - 16i^2$ $\newline$ Again, replacing $i^2$ with $-1$ , we get: $\newline$ $16 - 16i + 16i + 16 = 32$
Simplify Numerator and Denominator: Now we have the simplified numerator and denominator: $\newline$ $(-4 + 28i) / 32$ $\newline$ We can simplify this by dividing both the real part and the imaginary part by $32$ : $\newline$ $-\frac{4}{32} + \frac{28i}{32}$
Final Simplification: Simplify the fractions: $\newline$ $-\frac{1}{8} + \left(\frac{7}{8}\right)i$ $\newline$ This is the complex number in $a+bi$ form.

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