[math]4i\cdot3i[/math]

[math]4i\cdot\text{-}3i[/math]

[math]\text{-}2i\cdot\text{-}5i[/math]

[math]\text{-}5i\cdot5i[/math]

[math](\text{-}5i)^2[/math]

[math](\text{-}3+9i)(5i)[/math]

[math](8+i)(\text{-}5+3i)[/math]

[math](3+2i)^2[/math]

[math](3+2i)(3-2i)[/math]

On October 16, 1843, while walking across the Broom Bridge in Dublin, Ireland, Sir William Rowan Hamilton came up with an idea for numbers that would work sort of like complex numbers. Instead of just the number [math]i[/math] (and its opposite[math]-i[/math]) squaring to give -1, he imagined three numbers [math]i[/math], [math]j[/math], and [math]k[/math] (each with an opposite) that squared to give -1.[br][br]The way these numbers multiplied with each other was very interesting. [math]i[/math] times [math]j[/math] would give [math]k[/math], [math]j[/math] times [math]k[/math] would give [math]i[/math], and [math]k[/math] times [math]i[/math] would give [math]j[/math]. But the multiplication he imagined did not have a commutative property. When those numbers were multiplied in the opposite order, they’d give the opposite number. So [math]j[/math] times [math]i[/math] would give [math]\text{-}k[/math], [math]k[/math] times [math]j[/math] would give [math]\text{-}i[/math], and [math]i[/math] times [math]k[/math] would give [math]\text{-}j[/math]. A [i]quaternion[/i] is a number that can be written in the form [math]a+bi+cj+dk[/math] where [math]a[/math], [math]b[/math], [math]c[/math], and [math]d[/math] are real numbers.[br][br]Let [math]w=2+3i-j[/math] and [math]z=2i+3k[/math]. Write each given expression in the form [math]a+bi+cj+dk[/math].[br][br][math]w+z[/math]

[math]wz[/math]

[math]zw[/math]

[size=150]Which expression is equivalent to [math]2i(5+3i)[/math]?[/size]

[size=150]Lin says, “When you add or multiply two complex numbers, you will always get an answer you can write in [math]a+bi[/math] form.”[br][br]Noah says, “I don’t think so. Here are some exceptions I found:”[/size][br][br][math](7+2i)+(3-2i)=10[/math][br][br][math](2+2i)(2+2i)=8i[/math][br][br]Check Noah’s arithmetic. Is it correct?[br]

Can Noah’s answers be written in the form [math]a+bi[/math], where [math]a[/math] and [math]b[/math] are real numbers? Explain or show your reasoning.[br]

Explain to someone who missed class how you would write [math](3-5i)(\text{-}2+4i)[/math] in the form [math]a+bi[/math], where [math]a[/math] and [math]b[/math] are real numbers.

Which expression is equal to [math]729^{\frac{2}{3}}[/math]?

[math]2x^2-\frac{2}{3}=5\frac{1}{3}[/math]

[math](x+1)^2=81[/math]

[math]3x^2+14=12[/math]

Select [b]all[/b] the expressions that are equivalent to [math](3x+2)(x-4)[/math] for all real values of [math]x[/math].