heureka

if 3t-7= 5t then 6t=?

$$\begin{array}{rcl}
3t-7 &=& 5t \quad | \quad -3t\\
-7&=&2t\\
2t&=&-7 \quad | \quad *3\\
6t&=&-21
\end{array}$$

12m to the 7th power divided by 3m squared

$$\dfrac {(12m)^7}{(3m)^2}=\\\\
=\dfrac{12*12*12*12*12*12*12*m*m*m*m*m*\not{m}*\not{m}}{3*3*\not{m}*\not{m}}\\\\
=\dfrac{12*12*12*12*12*12*12*m*m*m*m*m}{3*3}\\\\
=\dfrac{12*12*12*12*12*\not{3}*4*\not{3}*4*m*m*m*m*m}{\not{3}*\not{3}}\\\\
=\dfrac{12*12*12*12*12*4*4*m*m*m*m*m}{1}\\$$

$$=12^5*4^2*m^5$$         |           $${{\mathtt{12}}}^{{\mathtt{5}}}{\mathtt{\,\times\,}}{{\mathtt{4}}}^{{\mathtt{2}}} = {\mathtt{3\,981\,312}}$$

$$\dfrac {(12m)^7}{(3m)^2}=3\;981\;312\;m^5$$

u= (1, 2, 3)    

v= (-2, 0, 1) 

$$\vec{w}=\vec{u}+\vec{v}
=
\left(
\begin{array}{c}
1 & 2 & 3
\end{array}
\right)
+\left(
\begin{array}{c}
-2 & 0 & 1
\end{array}
\right)
=
\left(
\begin{array}{c}
1-2 & 2+0 & 3+1
\end{array}
\right)
=
\left(
\begin{array}{c}
-1 & 2 & 4
\end{array}
\right)$$

$$\begin{array}{rcl}
\|w\| &=& \sqrt{(-1)^2+2^2+4^2} \\\\
&=&\sqrt{1+4+16}\\\\
&=&\sqrt{21}\\\\
& \approx & 4.58257569496
\end{array}$$

what is 15 % of 32 ?

$$\begin{array}{rclclcl}
1\% &of& 32 \quad = \dfrac{32}{100}&=&&&0.32\\\\
15\%&of& 32 &=&15&\times& 0.32\quad = \quad 4.8\\
\end{array}$$

$$a=4\quad b=6\quad c=5 \quad \mbox{ angle } \theta\mbox{ ? }$$

$$\begin{array}{l}
c^2=a^2+b^2-2ab*\cos{(\theta) } \\\\
\cos{(\theta)} = \dfrac{a^2+b^2-c^2}{2ab}\\\\
\cos{(\theta)} = \dfrac{4^2+6^2-5^2}{2*4*6}\\\\
\cos{(\theta)} = \dfrac{16+36-25}{48}\\\\
\cos{(\theta)} = \dfrac{27}{48}\\\\
\cos{(\theta)} = 0.5625\\\\
\textcolor[rgb]{1,0,0}{\theta=55.7711336722\ensuremath{^\circ}}
\end{array}$$

$$\begin{array}{lcl}
\dfrac{18}{72}&\times&\dfrac{100}{100}\\ \\
=\dfrac{18}{72}&\times&100*\dfrac{1}{100}\\\\
=\dfrac{18}{72}&\times&100*\%\\\\
=\dfrac{1800}{72}&\times& \% \\ \\
=\dfrac{1800}{72}\%\\\\
=25\%
\end{array}\\$$

$$4x^{\textcolor[rgb]{1,0,0}{8}}+2x^{ \textcolor[rgb]{0,0,1}{1}}-13$$

the polynomial degree  is 8 (The biggest exponent)

Hi Strider,

$$\alpha = 47\ensuremath{^\circ}\quad b=7\quad c=3.59\\
\mbox{side a?} \quad \mbox{angle } \beta \mbox{ ? and angle }\gamma \mbox{ ?}$$

a:

$$\begin{array}{l}a^2=b^2+c^2-2bc*\cos{(47\ensuremath{^\circ})} \\
a^2=7^2+3.59^2-2*7*3.59*\cos{(47\ensuremath{^\circ})} \\
a^2=27.6108624233\\
\textcolor[rgb]{1,0,0}{a=5.25460392639}
\end{array}$$

$$\mathbf{\beta}:$$

$$\begin{array}{l}b^2=a^2+c^2-2ac*\cos{(\beta) } \\
\cos{(\beta)} = \dfrac{a^2+c^2-b^2}{2ac}\\
\cos{(\beta)} = -0.22532402766\\
\textcolor[rgb]{1,0,0}{\beta=103.021932880\ensuremath{^\circ}}
\end{array}$$

$$\mathbf{\gamma}:$$

$$\begin{array}{l}c^2=a^2+b^2-2ab*\cos{(\gamma) } \\
\cos{(\gamma)} = \dfrac{a^2+b^2-c^2}{2ab}\\
\cos{(\gamma)} = 0.86621674081\\
\textcolor[rgb]{1,0,0}{\gamma=29.9780671201\ensuremath{^\circ}}
\end{array}$$

$$\boxed{\alpha + \beta +\gamma =
47\ensuremath{^\circ}
+103.021932880\ensuremath{^\circ}
+29.9780671201\ensuremath{^\circ}
=180.000000000\ensuremath{^\circ}}$$

$$\begin{array}{rcl}
vx^2&=&v^2-g^2t^2\\
\sqrt{vx^2}&=&\sqrt{v^2-g^2t^2} \\
\textcolor[rgb]{1,0,0}{\sqrt{v}}x&=&\sqrt{v^2-g^2t^2}
\end{array}$$

$$\boxed{\begin{array}{lrcrcrr} (1) &5x & - & 14 &=& a &\\
(2)& 2x& + & 1 &=& a &\\
&--&-&--&-&--&\\
(1) - (2)& 3x&-&15&=&0 & \quad | \quad :3\\
&x&-&5&=&0 & \quad | \quad +5\\
&\textcolor[rgb]{1,0,0}{x}&&&\textcolor[rgb]{1,0,0}{=}&\textcolor[rgb]{1,0,0}{5} &\end{array}}$$