# Algebra

## Real Numbers

Commutative
```
a + b = b + a
```

Associative
```
a + (b + c) = (a + b) + c
```

Distributive
```
(a + b)c = ac + bc
```

|Operation|Commutative|Associative|Distributive|
|-|-|-|-|
|Addition|✓|✓||
|Multiplication|✓|✓||

----

## Law of Exponents

$\displaystyle \begin{equation} \label{eq1}
\begin{split}
a^0 & = 1 \\
(a^m)(a^n) & = a^{m+n} \\
(ab)^m & = a^mb^m \\
(a^m)^n & = a^{mn} \\
\frac{a^m}{a^n} & = a^{m-n} \\
a^m & = \frac{1}{a^{-m}} \\
a^{-m} & = \frac{1}{a^{m}} \\
a^b & = e^{ln(a)b} \\
\end{split}
\end{equation}$

----

## Formulas
https://byjus.com/algebra-formulas

$\displaystyle \begin{equation} \label{eq1}
\begin{split}
a^2 - b^2 & = (a+b)(a-b) & \\
a^2 + 2ab + b^2 & = (a+b)^2 & \\
a^2 + b^2 & = (a-b)^2\;+ && 2ab \\
(a + b + c)^2 & = & a^2\;+\;b^2\;+\;c^2\;+\;& 2ab\;+\;& 2ac\;+\;& 2bc \\
(a\;–\;b\;–\;c)^2 & = & a^2\;+\;b^2\;+\;c^2\;–\;& 2ab\;–\;& 2ac\;+\;& 2bc \\
\end{split}
\end{equation}$

<br/>

$\displaystyle \begin{equation} \label{eq2}
\begin{split}
(a + b)^3 & = & a^3\;+\;& 3a^2b\;+\;& 3ab^2\;+\;& b^3 \\
 & = 3ab(a + b) & a^3\;+\;&&& b^3 \\
(a – b)^3 & = & a^3 \;–\;& 3a^2b\;+\;& 3ab^2\;–\;& b^3 \\
a^3 – b^3 & = (a – b)(a^2 + ab + b^2) & \\
a^3 + b^3 & = (a + b)(a^2 – ab + b^2) & \\
(a + b)^3 & = & a^3\;+\;& 3a^2b\;+\;& 3ab^2\;+\;& b^3 \\
(a – b)^3 & = & a^3\;–\;& 3a^2b\;+\;& 3ab^2\;-\;& b^3 \\
\end{split}
\end{equation}$

<br/>

$\displaystyle \begin{equation} \label{eq3}
\begin{split}
(a + b)^4 & = & a^4\;+\;& 4a^3b\;+\;& 6a^2b^2\;+\;& 4ab^3\;+\;& b^4 \\
(a – b)^4 & = & a^4\;–\;& 4a^3b\;+\;& 6a^2b^2\;–\;& 4ab^3\;+\;& b^4 \\
a^4 – b^4 & = (a – b)(a + b)(a^2 + b^2) & \\
\end{split}
\end{equation}$

<br/>

$\displaystyle \begin{equation} \label{eq4}
\begin{split}
a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4) \\
\end{split}
\end{equation}$






```python

```

e8564fddf376ed57b96a9eef3ae653bd8a1eb0d5
ipynb
Jupyter Notebook
notebooks/math/algebra/algebra.ipynb
sparkboom/my_jupyter_notes
9255e4236b27f0419cdd2c8a2159738d8fc383be
MIT
notebooks/math/algebra/algebra.ipynb
sparkboom/my_jupyter_notes
9255e4236b27f0419cdd2c8a2159738d8fc383be
MIT
notebooks/math/algebra/algebra.ipynb
sparkboom/my_jupyter_notes
9255e4236b27f0419cdd2c8a2159738d8fc383be
MIT
Qwen/Qwen-72B
1. YES
2. YES