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

(30-10 i)/(-10+10 i)
Answer:

Express as a complex number in simplest a+bi form: $\newline$ $\frac{30-10 i}{-10+10 i}$ $\newline$ Answer:

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Rational root theorem

Full solution

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

\newline

\frac{30-10 i}{-10+10 i}

\newline

Answer:

Write Given Complex Fraction: Write down the given complex fraction. $\newline$ We are given the complex fraction $(30-10i)/(-10+10i)$ and we need to express it in the form $a+bi$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ The conjugate of the denominator $-10+10i$ is $-10-10i$ . We multiply both the numerator and the denominator by this conjugate to remove the imaginary part from the denominator. $\newline$ $\frac{(30-10i)(-10-10i)}{(-10+10i)(-10-10i)}$
Multiply Numerator: Perform the multiplication in the numerator. $\newline$ $(30-10i)(-10-10i) = 30(-10) + 30(-10i) - 10i(-10) - 10i(-10i)$ $\newline$ $= -300 - 300i + 100i + 100$ $\newline$ $= -200 - 200i + 100i^2$ $\newline$ Since $i^2 = -1$ , we have: $\newline$ $= -200 - 200i - 100$ $\newline$ $= -300 - 200i$
Multiply Denominator: Perform the multiplication in the denominator. $\newline$ $(-10+10i)(-10-10i) = (-10)(-10) - 10i(-10) + 10i(-10) - (10i)(10i)$ $\newline$ $= 100 + 100i - 100i - 100i^2$ $\newline$ Since $i^2 = -1$ , we have: $\newline$ $= 100 + 100i - 100i + 100$ $\newline$ $= 200$
Divide by Denominator: Divide the numerator by the denominator. $\newline$ We have the numerator as $-300 - 200i$ and the denominator as $200$ . Dividing the numerator by the denominator gives us: $\newline$ $(-300 - 200i) / 200$ $\newline$ $= -300/200 - (200i/200)$ $\newline$ $= -1.5 - i$

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