Multiple Choice
Multiply the following and simplify.
9. Roots, Radicals, and Complex Numbers
Multiplying and Dividing Complex Numbers
Multiple Choice
Find the quotient. Express your answer in standard form.
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Verified step by step guidance1
Identify the problem: You need to divide the complex number \(6 + i\) by the complex number \(4 - 2i\) and express the result in standard form, which is \(a + bi\) where \(a\) and \(b\) are real numbers.
Multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. The conjugate of \(4 - 2i\) is \(4 + 2i\). So, multiply both numerator and denominator by \(4 + 2i\):
\[\frac{6 + i}{4 - 2i} \times \frac{4 + 2i}{4 + 2i}\]
Use the distributive property (FOIL) to expand both the numerator and the denominator:
- Numerator: \((6 + i)(4 + 2i)\)
- Denominator: \((4 - 2i)(4 + 2i)\)
Simplify the denominator using the difference of squares formula for complex numbers: \((a - bi)(a + bi) = a^2 + b^2\). Then, simplify the numerator by combining like terms (real and imaginary parts). Finally, write the quotient in the form \(a + bi\) by dividing the simplified numerator by the simplified denominator.
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