Write (4+5i)^(2) in simplest a+bi form. Answer:

Square Complex Number: To find the simplest a+bi form of (4+5i)^(2), we need to square the complex number (4+5i). We use the formula (a+bi)^2 = a^2 + 2abi + (bi)^2. Let's calculate it: (4+5i)^2 = 4^2 + 2*4*5i + (5i)^2.Calculate Real Part: First, we calculate the real part of the expression: 4^2 = 16.Calculate Imaginary Part: Next, we calculate the imaginary part without the i squared: 2*4*5 = 40, so the imaginary part is 40i.Calculate i Squared Part: Now, we calculate the i squared part: (5i)^2 = 25i^2. Since i^2 = -1, we have 25i^2 = 25*(-1) = -25.Combine Real and Imaginary Parts: We combine the real part and the imaginary parts: 16 (from step 2) + 40i (from step 3) - 25 (from step 4). So, (4+5i)^2 = 16 + 40i - 25.Final Simplified Form: Finally, we combine the real parts and keep the imaginary part separate: (16 - 25) + 40i = -9 + 40i. This is the simplest a+bi form of (4+5i)^(2).

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Q. Write

(4+5 i)^{2}

in simplest

a+b i

form.

\newline

Answer:

Square Complex Number: To find the simplest $a+bi$ form of $(4+5i)^{2}$ , we need to square the complex number $(4+5i)$ . We use the formula $(a+bi)^2 = a^2 + 2abi + (bi)^2$ . Let's calculate it: $(4+5i)^2 = 4^2 + 2\cdot4\cdot5i + (5i)^2$ .
Calculate Real Part: First, we calculate the real part of the expression: $4^2 = 16$ .
Calculate Imaginary Part: Next, we calculate the imaginary part without the $i^2$ : $2 \times 4 \times 5 = 40$ , so the imaginary part is $40i$ .
Calculate i Squared Part: Now, we calculate the i squared part: $(5i)^2 = 25i^2$ . Since $i^2 = -1$ , we have $25i^2 = 25*(-1) = -25$ .
Combine Real and Imaginary Parts: We combine the real part and the imaginary parts: $16$ (from step $2$ ) + $40i$ (from step $3$ ) - $25$ (from step $4$ ). $\newline$ So, $(4+5i)^2 = 16 + 40i - 25$ .
Final Simplified Form: Finally, we combine the real parts and keep the imaginary part separate: $16 - 25$ + $40i$ = $-9 + 40i$ . This is the simplest $a+bi$ form of $4+5i$ ^{ $2$ }.