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

(-10+6i)/(5-7i)
Answer:

Express as a complex number in simplest a+bi form:\newline10+6i57i \frac{-10+6 i}{5-7 i} \newlineAnswer:

Full solution

Q. Express as a complex number in simplest a+bi form:\newline10+6i57i \frac{-10+6 i}{5-7 i} \newlineAnswer:
  1. Multiply Numerators: To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of (57i)(5-7i) is (5+7i)(5+7i).\newlineCalculation: Multiply (10+6i)(-10+6i) by the conjugate of the denominator (5+7i)(5+7i), and also multiply the denominator (57i)(5-7i) by its conjugate (5+7i)(5+7i).\newlineMath error check: No calculations have been made yet.
  2. Multiply Denominators: Multiplying the numerators: (10+6i)×(5+7i)=5070i+30i+42i2(-10+6i) \times (5+7i) = -50 - 70i + 30i + 42i^2. Since i2=1i^2 = -1, we can simplify this to 5070i+30i42-50 - 70i + 30i - 42.\newlineCalculation: 5040i42=9240i-50 - 40i - 42 = -92 - 40i.\newlineMath error check: No math error in this step.
  3. Divide Simplified Numbers: Multiplying the denominators: (57i)×(5+7i)=25+35i35i49i2(5-7i) \times (5+7i) = 25 + 35i - 35i - 49i^2. Since i2=1i^2 = -1, this simplifies to 2549(1)25 - 49(-1).\newlineCalculation: 25+49=7425 + 49 = 74.\newlineMath error check: No math error in this step.
  4. Simplify Fractions: Now we divide the simplified numerator by the simplified denominator to get the complex number in a+bia+bi form.\newlineCalculation: (9240i)/74=92/74(40i/74)(-92 - 40i) / 74 = -92/74 - (40i/74).\newlineMath error check: No math error in this step.
  5. Simplify Fractions: Now we divide the simplified numerator by the simplified denominator to get the complex number in a+bia+bi form.\newlineCalculation: (9240i)/74=92/74(40i/74)(-92 - 40i) / 74 = -92/74 - (40i/74).\newlineMath error check: No math error in this step. Simplify the fractions: 92/74-92/74 simplifies to 46/37-46/37 and 40i/74-40i/74 simplifies to 20i/37-20i/37.\newlineCalculation: 46/3720i/37-46/37 - 20i/37.\newlineMath error check: No math error in this step.

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