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Express as a complex number in simplest a+bi form: (6-6i)/(2-i) Answer:

Express as a complex number in simplest a+bi form:\newline6−6i2−i \frac{6-6 i}{2-i} \newlineAnswer:

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Full solution

Q. Express as a complex number in simplest a+bi form:\newline6−6i2−i \frac{6-6 i}{2-i} \newlineAnswer:
  1. Write Given Complex Fraction: Write down the given complex fraction.\newlineWe have the complex fraction (6−6i)/(2−i)(6-6i)/(2-i) that we want to express in the form a+bia+bi.
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of (2−i)(2-i) is (2+i)(2+i). We multiply both the numerator and the denominator by this conjugate to eliminate the imaginary unit ii from the denominator. (6−6i)(2+i)(2−i)(2+i)\frac{(6-6i)(2+i)}{(2-i)(2+i)}
  3. Perform Multiplication: Perform the multiplication in the numerator and the denominator.\newlineNumerator: (6−6i)(2+i)=12+6i−12i−6i2(6-6i)(2+i) = 12+6i-12i-6i^2\newlineDenominator: (2−i)(2+i)=4+2i−2i−i2(2-i)(2+i) = 4+2i-2i-i^2
  4. Simplify Expressions: Simplify the expressions in the numerator and the denominator.\newlineNumerator: 12+6i−12i−6i2=12−6i+612+6i-12i-6i^2 = 12-6i+6 (since i2=−1i^2 = -1)\newlineDenominator: 4+2i−2i−i2=4+14+2i-2i-i^2 = 4+1 (since i2=−1i^2 = -1)
  5. Combine Like Terms: Combine like terms in the numerator and the denominator.\newlineNumerator: 12−6i+6=18−6i12-6i+6 = 18-6i\newlineDenominator: 4+1=54+1 = 5
  6. Divide Numerator by Denominator: Divide the numerator by the denominator to get the complex number in a+bia+bi form.\newline(18−6i)/5=18/5−(6i/5)(18-6i)/5 = 18/5 - (6i/5)
  7. Write Final Answer: Write the final answer in a+bia+bi form.\newlineThe complex number in a+bia+bi form is 185−65i\frac{18}{5} - \frac{6}{5} i, or 3.6−1.2i3.6 - 1.2i.

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