$$ 
\sqrt{2}+\sqrt{3}=\sqrt{\left(\sqrt{2}+\sqrt{3}\right)^2}=\sqrt{2\sqrt{6}+5}
=\sqrt{\sqrt{\left(2\sqrt{6}+5\right)^2}} = \sqrt{\sqrt{20\sqrt{6}+49}}
$$


```python
import sympy as S
S.init_printing()
a = S.sqrt( S.sqrt(49+20*S.sqrt(6)))

a
```


```python
S.sqrtdenest(a)
```

$$ 
\sqrt{2}+\sqrt{3}=\sqrt{2\sqrt{6}+5}=\sqrt{1+(2\sqrt{6}+4)}
=\sqrt{1+\sqrt{\left(2\sqrt{6}+4\right)^2}} = \sqrt{1+\sqrt{16\sqrt{6}+40}}
$$


```python
b = S.sqrt(1+S.sqrt(40+16*S.sqrt(6)))

b
```


```python
S.sqrtdenest(b)
```

bfbf37d8ec53794a41d896a7ab159d1e9e10d977
ipynb
Jupyter Notebook
sympy_sqrtdenest.ipynb
hamukazu/notebook-misc
1b39d137f99dcf0495dc101f82997e669ff6dead
MIT
sympy_sqrtdenest.ipynb
hamukazu/notebook-misc
1b39d137f99dcf0495dc101f82997e669ff6dead
MIT
sympy_sqrtdenest.ipynb
hamukazu/notebook-misc
1b39d137f99dcf0495dc101f82997e669ff6dead
MIT
Qwen/Qwen-72B
1. YES
2. YES