Multiple Choice
Find the product. Express your answer in standard form.
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9. Roots, Radicals, and Complex Numbers
Multiplying and Dividing Complex Numbers
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Find the product of the given complex number and its conjugate.
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Verified step by step guidance1
Identify the given complex number, which is \$4 - 5i\(, and write down its conjugate. The conjugate of a complex number \)a + bi\( is \)a - bi\(, so the conjugate of \)4 - 5i\( is \)4 + 5i$.
Set up the product of the complex number and its conjugate: \((4 - 5i)(4 + 5i)\).
Use the distributive property (FOIL method) to expand the product: multiply the first terms, outer terms, inner terms, and last terms.
Recall that \(i^2 = -1\), so when you multiply the imaginary parts, replace \(i^2\) with \(-1\) to simplify the expression.
Combine like terms (real parts and imaginary parts) to get the final simplified product, which will be a real number.
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