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Express as a complex number in simplest a+bi form: (22-24 i)/(2-7i) Answer:

Express as a complex number in simplest a+bi form:\newline22−24i2−7i \frac{22-24 i}{2-7 i} \newlineAnswer:

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Full solution

Q. Express as a complex number in simplest a+bi form:\newline22−24i2−7i \frac{22-24 i}{2-7 i} \newlineAnswer:
  1. Multiply by Conjugate: To express the complex fraction (22−24i)/(2−7i)(22-24i)/(2-7i) in the form a+bia+bi, we need to eliminate the imaginary part from the denominator. We do this by multiplying the numerator and the denominator by the complex conjugate of the denominator.\newlineThe complex conjugate of (2−7i)(2-7i) is (2+7i)(2+7i).\newlineNow, multiply the numerator and the denominator by (2+7i)(2+7i).
  2. Numerator Multiplication: Perform the multiplication in the numerator: (22−24i)×(2+7i)(22-24i) \times (2+7i). This involves using the distributive property (FOIL method) to expand the product. (22×2)+(22×7i)+(−24i×2)+(−24i×7i)(22\times2) + (22\times7i) + (-24i\times2) + (-24i\times7i) = 44+154i−48i−168i244 + 154i - 48i - 168i^2 Since i2=−1i^2 = -1, replace −168i2-168i^2 with 168168. = 44+106i+16844 + 106i + 168 = 212+106i212 + 106i
  3. Denominator Multiplication: Perform the multiplication in the denominator: (2−7i)×(2+7i)(2-7i) \times (2+7i). This is a difference of squares, which simplifies to: (22)−(7i)2=4−49i2(2^2) - (7i)^2 = 4 - 49i^2 Since i2=−1i^2 = -1, replace −49i2-49i^2 with 4949. =4+49=53= 4 + 49 = 53
  4. Divide and Simplify: Now, divide the simplified numerator by the simplified denominator to get the complex number in a+bia+bi form.\newline(212+106i)/53(212 + 106i) / 53\newlineDivide both the real part and the imaginary part by 5353.\newline21253+(10653)i\frac{212}{53} + \left(\frac{106}{53}\right)i\newline= 4+2i4 + 2i

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