((1)/(2)-(5)/(2)i)((1)/(3)+(2)/(3)i)

Multiply Real Parts: Multiply the real parts of the complex numbers. (1/2) * (1/3) = 1/6Multiply Imaginary Parts: Multiply the imaginary parts of the complex numbers and include the negative sign from the first complex number. (-5/2i) * (2/3i) = -10/6 * i^2 Since i^2 = -1, we have -10/6 * (-1) = 10/6, which simplifies to 5/3.Multiply Mixed Parts: Multiply the real part of the first complex number with the imaginary part of the second complex number. (1/2) * (2/3i) = 1/3iCombine Results: Multiply the imaginary part of the first complex number with the real part of the second complex number. (-5/2i) * (1/3) = -5/6iWrite Final Answer: Combine the results from steps 1, 2, 3, and 4. Real parts: 1/6 + 5/3 = 1/6 + 10/6 = 11/6 Imaginary parts: 1/3i - 5/6i = 2/6i - 5/6i = -3/6i, which simplifies to -1/2iWrite the final answer as a complex number. The real part is 11/6 and the imaginary part is -1/2i. So, the product of the complex numbers is 11/6 - 1/2i.

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$\left(\frac{1}{2}-\frac{5}{2}i\right)\left(\frac{1}{3}+\frac{2}{3}i\right)$

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Algebra 1

Multiplication with rational exponents

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\left(\frac{1}{2}-\frac{5}{2}i\right)\left(\frac{1}{3}+\frac{2}{3}i\right)

Multiply Real Parts: Multiply the real parts of the complex numbers. $\newline$ $(\frac{1}{2}) \times (\frac{1}{3}) = \frac{1}{6}$
Multiply Imaginary Parts: Multiply the imaginary parts of the complex numbers and include the negative sign from the first complex number. $\newline$ $(-\frac{5}{2}i) \times (\frac{2}{3}i) = -\frac{10}{6} \times i^2$ $\newline$ Since $i^2 = -1$ , we have $-\frac{10}{6} \times (-1) = \frac{10}{6}$ , which simplifies to $\frac{5}{3}$ .
Multiply Mixed Parts: Multiply the real part of the first complex number with the imaginary part of the second complex number. $\newline$ $(\frac{1}{2}) \times (\frac{2}{3}i) = \frac{1}{3}i$
Combine Results: Multiply the imaginary part of the first complex number with the real part of the second complex number. $(-\frac{5}{2}i) \times (\frac{1}{3}) = -\frac{5}{6}i$
Write Final Answer: Combine the results from steps $1$ , $2$ , $3$ , and $4$ . $\newline$ Real parts: $\frac{1}{6} + \frac{5}{3} = \frac{1}{6} + \frac{10}{6} = \frac{11}{6}$ $\newline$ Imaginary parts: $\frac{1}{3}i - \frac{5}{6}i = \frac{2}{6}i - \frac{5}{6}i = -\frac{3}{6}i$ , which simplifies to $-\frac{1}{2}i$
Write Final Answer: Combine the results from steps $1$ , $2$ , $3$ , and $4$ . $\newline$ Real parts: $\frac{1}{6} + \frac{5}{3} = \frac{1}{6} + \frac{10}{6} = \frac{11}{6}$ $\newline$ Imaginary parts: $\frac{1}{3}i - \frac{5}{6}i = \frac{2}{6}i - \frac{5}{6}i = -\frac{3}{6}i$ , which simplifies to $-\frac{1}{2}i$ Write the final answer as a complex number. $\newline$ The real part is $\frac{11}{6}$ and the imaginary part is $-\frac{1}{2}i$ . $\newline$ So, the product of the complex numbers is $\frac{11}{6} - \frac{1}{2}i$ .