Express (3/2 + √5/4 i)^2 in simpest a + bi form

simplest* oops

Does it look like this?

$({\frac{3}{2} + \frac{i\sqrt{5}}{4}})^{2}$

If it does, then:

$({\frac{3}{2} + \frac{i\sqrt{5}}{4}})({\frac{3}{2} + \frac{i\sqrt{5}}{4}})$

Foil it out:

$(\frac{3}{2}\times\frac{3}{2}) + (\frac{3}{2}\times\frac{i\sqrt{5}}{4}) + (\frac{3}{2}\times\frac{i\sqrt{5}}{4}) + (\frac{i\sqrt{5}}{4}\times\frac{i\sqrt{5}}{4})$

Simplify the multiplication parts:

$\frac{6}{4} + \frac{3i\sqrt{5}}{4} + \frac{3i\sqrt{5}}{4} - \frac{5}{16}$

Combine like terms:

$\frac{24}{16} + \frac{6i\sqrt{5}}{4} - \frac{5}{16}$

Combine more like terms:

$\frac{19}{16} + \frac{3i\sqrt{5}}{2}$

And there you have it. I hope I understood the original question correctly.

Expand the following:
(3/2+(sqrt(5))/(4) i)^2

(3/2+(sqrt(5) i)/(4))^2 = 9/4+1/8 (3 i) sqrt(5)+1/8 (3 i) sqrt(5)-5/16 = 31/16+1/4 (3 i) sqrt(5):
Answer:  31/16+(3 i)/4 sqrt(5)= 1.9375+1.677051i

Express (3/2 + √5/4 i)^2 in simpest a + bi form

Binom: $\begin{array}{rcll} \left( \frac32 + \frac{ \sqrt{5} } {4} i \right)^2 \end{array}$

$\begin{array}{rcll} \left( \frac32 + \frac{ \sqrt{5} } {4} i \right)^2 &=& \left( \frac32 \right)^2 + 2\cdot \frac32 \cdot \frac{ \sqrt{5} } {4} i + \left( \frac{ \sqrt{5} } {4} i \right)^2 \\ &=& \frac94 + 3 \cdot \frac{ \sqrt{5} } {4} i + \frac{ 5 } {16} i^2 \qquad | \qquad i^2 = -1\\ &=& \frac94 + 3 \cdot \frac{ \sqrt{5} } {4} i + \frac{ 5 } {16} \cdot( -1 )\\ &=& \frac94 + 3 \cdot \frac{ \sqrt{5} } {4} i - \frac{ 5 } {16}\\ &=& \frac94 - \frac{ 5 } {16} + 3 \cdot \frac{ \sqrt{5} } {4} i \\ &=& \frac{9\cdot4}{4\cdot 4} - \frac{ 5 } {16} + 3 \cdot \frac{ \sqrt{5} } {4} i \\ &=& \frac{36}{16} - \frac{ 5 } {16} + 3 \cdot \frac{ \sqrt{5} } {4} i \\ &=& \frac{36-5}{16} + 3 \cdot \frac{ \sqrt{5} } {4} i \\ &=& \frac{31}{16} + 3 \cdot \frac{ \sqrt{5} } {4} i \\ &=& \frac{31}{16} + \frac34 \cdot \sqrt{5} \cdot i \\ &=& 1.9375 + 1.67705098312 \cdot i \\ \end{array}$