heureka

$$\small{\text{Formel:$\qquad \boxed{\left( \dfrac{a}{b} \right)^{-n}= \left( \dfrac{b}{a} \right)^{n}}$}}\\\\\\
\left( \dfrac{e}{ 1 } \right)^{-2}=\left( \dfrac{1}{ e } \right)^{2} =\dfrac{1}{e^2}$$

You have your alignment as rcl   right, centre, left  but there are no & symbols.

I experimented with leaving out the cl  and it looked just the same - do they do anything?

they do nothing, only placeholder!

What is  \text{$   for?   What does it do, I see no text.

big         \small{mathe-Modus}}     :             $$\small{mathe-Modus}}$$

small      \small{\text{text-Modus}} :            $$\small{\text{text-Modus}}$$

small - italic    \small{\text{$mathe-Modus-italic$}} :     $$\small{\text{$mathe-Modus-italic$}}$$

I have trouble remembering \cdot because it means nothing to me.  Do you think the c stands for centre ?

Are there any other types of dots that you can have?

horizontal, center (c): \cdots      $$A_{11}\cdots A_{1n}$$

horizontal, down:        \ldots      $$A \ldots A$$

Example:

_pF_q(a_1, \ldots, a_p; c_1, \ldots, c_q; z) =
\sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(c_1)_n \cdots (c_q)_n} \frac{z^n}{n!}

$$_pF_q(a_1, \ldots, a_p; c_1, \ldots, c_q; z) =
\sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(c_1)_n \cdots (c_q)_n} \frac{z^n}{n!} \,$$

diagonal (d) :     \ddots      $$\ddots$$

vertical(v) :       \vdots       $$\vdots$$

Dots

The most common dot symbols used in math notation are available in LaTeX as well.

double (d):        \ddot       $$\ddot{a}$$

single:               \dot       $$\dot{a}$$

formula for sin^2(x) ??

$$\mathbf{
\sin^2{x} = \frac{1}{2}\ \Big(1 - \cos (2x) \Big)
}$$

Use two unit multipliers to convert 1000 mm² to square centimeters.

$$\small{\text{$
\begin{array}{lcl}
1000~mm^2\cdot \left( \dfrac{1~cm}{10~mm} \right)\cdot \left( \dfrac{1~cm}{10~mm} \right) \\\\
&=&\dfrac{1000}{10\cdot 10} ~cm^2\\\\
&=&\dfrac{1000}{100} ~cm^2\\\\
&=&10 ~cm^2\\\\
\end{array}
$}}$$

Drücke zuerst den $$2^{nd}$$ Button und anschließend den % Button

A sample of microbes weighs 50 mg. There are approximately 2.5 × 10⁶ microbes in the sample. What is the approximate mass of each microbe expressed using scientific notation in grams.

$$\small\text{$
\begin{array}{lcl}
\mathbf{\dfrac{50~mg}{2.5\cdot 10^6~microbes} \cdot \dfrac{1~g}{1000~mg} } \\\\
\mathbf{ =\dfrac{50}{2.5\cdot 10^6\cdot 1000} \cdot \dfrac{g}{ microbe } } \\\\
\mathbf{ =\dfrac{50}{2.5\cdot 10^6\cdot 10^3} \cdot \dfrac{g}{ microbe } } \\\\
\mathbf{ =\dfrac{50}{2.5\cdot 10^9 } \cdot \dfrac{g}{ microbe } } \\\\
\mathbf{ =\dfrac{50}{2.5 } \cdot 10^{-9} \cdot \dfrac{g}{ microbe } } \\\\
\mathbf{ =20 \cdot 10^{-9} \cdot \dfrac{g}{ microbe } } \\\\
\mathbf{ =20 \cdot \dfrac{ng}{ microbe } } \qquad ng = $ Nano gram$
\end{array}\\
$}}$$

the approximate mass of each microbe  is 20 ng (Nano gram) or $$\small{\text{$20 \cdot 10^{-9} ~g$}}$$

Simplify 12a³b² ÷ 4ab⁻².

$$\\\small{\text{
$
\begin{array}{rcl}
\dfrac{ 12\cdot a^3 \cdot b^2 }{ 4 \cdot a \cdot b^{-2} } \\\\
=\dfrac{ 12 } {4} \cdot \dfrac{ a^3 }{ a } \cdot \dfrac{ b^2 }{ b^{-2} } \\\\
=\dfrac{ 3\cdot 4 } {4} \cdot \dfrac{ a\cdot a^2 }{ a } \cdot \dfrac{ b^2 }{ b^{-2} } \\\\
=\dfrac{ 3\cdot \not{4} } {\not{4}} \cdot \dfrac{ \not{a}\cdot a^2 }{ \not{a} } \cdot \dfrac{ b^2 }{ b^{-2} } \\\\
= 3\cdot a^2 \cdot \dfrac{ b^2 }{ b^{-2} } \\\\
= 3\cdot a^2 \cdot b^{2-(-2) } \\\\
= 3\cdot a^2 \cdot b^{2+2} \\\\
= 3\cdot a^2 \cdot b^4
\end{array}
$}}$$