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Express as a complex number in simplest a+bi form: (16-28 i)/(-4-6i) Answer:

Express as a complex number in simplest a+bi form:\newline16−28i−4−6i \frac{16-28 i}{-4-6 i} \newlineAnswer:

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Full solution

Q. Express as a complex number in simplest a+bi form:\newline16−28i−4−6i \frac{16-28 i}{-4-6 i} \newlineAnswer:
  1. Identify complex conjugate: Identify the complex conjugate of the denominator.\newlineTo divide complex numbers, we multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of (−4−6i)(-4-6i) is (−4+6i)(-4+6i).
  2. Multiply by conjugate: Multiply the numerator and the denominator by the complex conjugate of the denominator.\newlineWe have (16−28i)/(−4−6i)(16-28i)/(-4-6i). Multiply by (−4+6i)/(−4+6i)(-4+6i)/(-4+6i) to get rid of the imaginary part in the denominator.
  3. Distribute and multiply numerators: Apply the distributive property to multiply out the numerators.\newlineMultiply (16−28i)(16-28i) by (−4+6i)(-4+6i).\newline(16−28i)(−4+6i)=16(−4)+16(6i)−28i(−4)−28i(6i)(16-28i)(-4+6i) = 16(-4) + 16(6i) - 28i(-4) - 28i(6i)
  4. Calculate products in numerators: Calculate the products in the numerators.\newline16(−4)=−6416(-4) = -64\newline16(6i)=96i16(6i) = 96i\newline−28i(−4)=112i-28i(-4) = 112i\newline−28i(6i)=−168i2-28i(6i) = -168i^2 (Remember that i2=−1i^2 = -1)
  5. Combine like terms in numerator: Combine like terms in the numerator.\newline−64+96i+112i−168(−1)-64 + 96i + 112i - 168(-1)\newline−64+208i+168-64 + 208i + 168\newline104+208i104 + 208i
  6. Distribute and multiply denominators: Apply the distributive property to multiply out the denominators.\newlineMultiply (−4−6i)(-4-6i) by (−4+6i)(-4+6i).\newline(−4−6i)(−4+6i)=(−4)(−4)+(−4)(6i)−6i(−4)−6i(6i)(-4-6i)(-4+6i) = (-4)(-4) + (-4)(6i) - 6i(-4) - 6i(6i)
  7. Calculate products in denominators: Calculate the products in the denominators.\newline(−4)(−4)=16(-4)(-4) = 16\newline(−4)(6i)=−24i(-4)(6i) = -24i\newline−6i(−4)=24i-6i(-4) = 24i\newline−6i(6i)=−36i2-6i(6i) = -36i^2 (Again, i2=−1i^2 = -1)
  8. Combine like terms in denominator: Combine like terms in the denominator.\newline16−24i+24i−36(−1)16 - 24i + 24i - 36(-1)\newline16+3616 + 36\newline5252
  9. Write in a+bia+bi form: Write the complex number in a+bia+bi form. Now we have the numerator 104+208i104 + 208i and the denominator 5252. Divide both the real part and the imaginary part of the numerator by the denominator. (104+208i)/52(104 + 208i) / 52 104/52+(208i/52)104/52 + (208i/52) 2+4i2 + 4i

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