Identify Terms: Identify the terms to be multiplied. We have the complex number 4i and the complex number (1/2) - (3/2)i that need to be multiplied together.Distribute Multiplication: Distribute the multiplication over the addition. We will use the distributive property to multiply 4i with each term in the second complex number. 4i * (1/2) - 4i * (3/2)iMultiply Real Part: Multiply the real part of the second complex number by 4i. Multiplying the real part (1/2) by 4i gives us: 4i * (1/2) = 2iMultiply Imaginary Part: Multiply the imaginary part of the second complex number by 4i. Multiplying the imaginary part (-3/2)i by 4i gives us: 4i * (-3/2)i = -6i^2Substitute i^2: Remember that i^2 is equal to -1. Substitute i^2 with -1 in the expression from Step 4. -6i^2 = -6(-1) = 6Combine Results: Combine the results from Step 3 and Step 5. Add the real part from Step 5 to the imaginary part from Step 3. 2i + 6Write Final Result: Write the final result in standard form. The final result is a complex number with a real part and an imaginary part. The real part is 6, and the imaginary part is 2i.

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$4i\left(\frac{1}{2}-\frac{3}{2}i\right)$

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Algebra 1

Multiplication with rational exponents

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4i\left(\frac{1}{2}-\frac{3}{2}i\right)

Identify Terms: Identify the terms to be multiplied. $\newline$ We have the complex number $4i$ and the complex number $(1/2) - (3/2)i$ that need to be multiplied together.
Distribute Multiplication: Distribute the multiplication over the addition. We will use the distributive property to multiply $4i$ with each term in the second complex number. $4i \times (\frac{1}{2}) - 4i \times (\frac{3}{2})i$
Multiply Real Part: Multiply the real part of the second complex number by $4i$ . Multiplying the real part $(1/2)$ by $4i$ gives us: $4i \times (1/2) = 2i$
Multiply Imaginary Part: Multiply the imaginary part of the second complex number by $4i$ . Multiplying the imaginary part $(-\frac{3}{2})i$ by $4i$ gives us: $4i \cdot (-\frac{3}{2})i = -6i^2$
Substitute $i^2$ : Remember that $i^2$ is equal to $-1$ . Substitute $i^2$ with $-1$ in the expression from Step $4$ . $-6i^2 = -6(-1) = 6$
Combine Results: Combine the results from Step $3$ and Step $5$ . $\newline$ Add the real part from Step $5$ to the imaginary part from Step $3$ . $\newline$ $2i + 6$
Write Final Result: Write the final result in standard form. $\newline$ The final result is a complex number with a real part and an imaginary part. $\newline$ The real part is $6$ , and the imaginary part is $2i$ .