No. exponentiation and square root are not algebraic operations. Only addition and multiplication are. That's kind of the whole point of this theory / subfield of mathematics.
The operations you mentioned can be found in many different subfields of mathematics, eg. real analysis, number theory, or even arithmetic (using a more broad, but a well-accepted definition). But not in algebra. It's the point of algebra to only have addition and multiplication. And it's why, for example, algebraic geometry exists (because algebraic geometers want to avoid transcendental functions like sqrt(), sin() etc.)
The operations you mentioned can be found in many different subfields of mathematics, eg. real analysis, number theory, or even arithmetic (using a more broad, but a well-accepted definition). But not in algebra. It's the point of algebra to only have addition and multiplication. And it's why, for example, algebraic geometry exists (because algebraic geometers want to avoid transcendental functions like sqrt(), sin() etc.)