Rom

\(a^2 + ab + b^2 = 0\\ (a+b)^2 = a^2+2ab+b^2= ab\\ a = \dfrac{-b\pm \sqrt{-3b^2}}{2} = b e^{\pm i 2\pi/3}\\ a^9+b^9 = b^9(e^{\pm i 6\pi}+1) = 2b^9\\ (a+b)^9 =( (a+b)^2)^{9/2} = \\ (ab)^{9/2} =\left( b^2 e^{\pm i 2\pi/3}\right)^{9/2} = b^9e^{\pm i 3\pi} = -b^9\\ \dfrac{a^9+b^9}{(a+b)^9} = \dfrac{2b^9}{-b^9} = -2\)

\(1) \text{Find vectors defining parallelogram from one vertex}\\ v_1=(1,1)-(0,2) = (1,-1)\\ v_2=(2,3)-(0,2) = (2,1)\\ 2) \text{augment these to 3D by setting the z coordinate to 0}\\ v_1= (1,-1,0)\\ v_2= (2,1,0)\\ 3) A\hat{k} = v_1 \times v_2\\ \begin{pmatrix}i&j&k\\1&-1&0\\2&1&0\end{pmatrix} = (0, 0,3) = 3\hat{k}\\ A=3\)

\(3\sqrt{18}+9\sqrt{50} = \\ 9\sqrt{2} + 45\sqrt{2} = \\ 54\sqrt{2}\)

\(T = \mu + z \sigma = 17 + (2.8)(4) = 28.2\\ P[\text{someone scores > Sarah}] = 0.00255513\\ \text{So Sarah's score is pretty exceptional}\)

\(f(x) = a x^3 + b x ^2 + c x + d\\ f(0) = 0 \Rightarrow d = 0\\ \begin{pmatrix}-1 &1 &-1 \\1 &1 &1 \\8 &4 &2\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix}= \begin{pmatrix}15 \\-5 \\12\end{pmatrix}\)

\(\text{Use Gaussian elimination to obtain}\\ \begin{pmatrix}a\\b\\c\end{pmatrix} = \begin{pmatrix}2 \\5\\-12\end{pmatrix}\\ f(x) =2x^3 +5x^2 -12x\)

\(6=2^13^1\\ \text{let }n = \prod \limits_{k=0}^\infty p_k^{n_k} \\ \text{where }p_k \text{ is the }kth \text{ prime}\\ 6n = 2^{n_2+1}\cdot 3^{n_3+1}\cdot \prod \limits_{k=3}^\infty p_k^{n_k} \)

\(6n \text{ thus has }(n_2+2)(n_3+2)\prod \limits_{k=0}^\infty (n_k+1) = 9 \text{ divisors}\)

\(\text{This is easily enough accomplished if }n_2=1,~n_3=1,~n_k=0,k>3, \text{ i.e. }\\ 6n=36 = 2^2 3^2\\ \text{There are a total of 2 prime divisors}\)

\(fv=15500\\ r=0.062\\ n=15\\ fv = pv \cdot e^{r t}\\ pv = fv \cdot e^{-r t} = 15500 \cdot e^{-15(0.062)} = \$6115.58\)

sacrilege!

\(a_n = \dfrac{\cos\left(\pi + \dfrac{2\pi n}{3}\right)}{\left|\cos\left(\pi + \dfrac{2\pi n}{3}\right)\right|}\)

figure a brick is 8x4x2 inches

3x6x3 = 54 bricks is 2 cubic feet

a fridge is about 6 foot high, 3 foot wide,  2 foot deep or 36 cubic feet.

36/2 = 12 x 54 = 648 bricks