algebra

Simplify 12a³b² ÷ 4ab⁻².

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

Another great answer Heureka :)

I have a couple of questions about your coding Heureka   

1)  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?

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

3)  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?

Thank you :)

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

\\\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}$}

Hi Heureka,

I added some questions for you that you may not have seen  (it was an edit)

If you get a chance could you take a look please?

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       $$\dot{a}$$

WOW there are a lot of dot commands.   Thanks Heureka.

wWhat about your used of  \text    There doesn't seem to be any text there so what is it for?

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

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!

Thank you Heureka