How to compute exponentiation and roots in the shell?

Question

I'm a beginner in shell and linux in general, it's a Shell Arithmetic question and I can't figure out what to write in terminal to solve these three equations. I'm sorry if it seems a bad question, I tried echo command and expr, but all of them are wrong and with many different errors like, '(' , syntax error near..., E0F, and many other unfortunately. I hope someone will provide me with the correct commands. And I appreciate any help from you. I'll put down the terminal codes that I used which are wrong as I know.

$ x=8
$ y=21
$ echo $((2*x**3 + sqrt(y/2))
bash: unexpected EOF while looking for matching ')'
bash: syntax error: unexpected end of file
$ echo $((2*x**3) + (sqrt(y/2)))
bash: command substitution: line 1: syntax error near unexpected token +'
bash: command substitution: line 1: `(2*x**3) + (sqrt(y/2))'
$ echo $((2*x**3)+(sqrt(y/2))
bash: unexpected EOF while looking for matching )'
bash: syntax error: unexpected end of file
$ echo $((2*x**3)+(sqrt(y/2)))
bash: command substitution: line 1: syntax error near unexpected token +(sqrt(y/2))'
bash: command substitution: line 1: `(2*x**3)+(sqrt(y/2))'
$ echo $((2x**3)+(sqrt(y / 2)))
bash: command substitution: line 1: syntax error near unexpected token +(sqrt(y / 2))'
bash: command substitution: line 1: (2x**3)+(sqrt(y / 2))'

Answer 1

The shell is not the right tool to do floating point computations. It only does integer math and does not provide functions like square root.

However, the bc utility does both. It is an arbitrary-precision decimal arithmetic language and calculator.

$ bc
>>> scale=5
>>> sqrt(21)
4.58257
>>> scale=19
>>> sqrt(21)
4.5825756949558400065
>>> x=8
>>> y=21
>>> x+5
13
>>> x^2
64
>>> 2*x^2 - sqrt(y/2)
124.7596296507960698846
>>> Type Control-D to exit interactive bc.
$

Be sure to read the manual page for bc with man bc to understand all of its capabilities and limitations.