Difference between calculator and google calc for power

Question

I tried to compute the power of 2^2^2^2 on google calculator and my casio calculator but both are giving different results. same is true for 3^3^3. Please explain me the difference between two expressions.

Answer 1

Your calculator is interpreting it as this:

$${{2^2}^2}^2 = ((2^2)^2)^2 = (4^2)^2 = 16^2 = 256$$

Google is interpreting it as this:

$${{2^2}^2}^2 = 2^{(2^{(2^2)})} = 2^{(2^4)} = 2^{16} = 65536$$

Similarly with the threes.

Technically Google is correct because order of operations says to do exponents first. So when we want to evaluate $2^{\color{red}{2}^{\color{blue}{2^2}}}$, order of operations says to evaluate the $\color{red}{{2^{\color{blue}{2^2}}}}$ first, i.e., evaluate the exponent first. Apply this rule again and it tells us we're supposed to evaluate the $\color{blue}{2^2}$ first, which is $4.$ Therefore $\color{red}{{2^{\color{blue}{2^2}}}} = 2^4 = 16$, and so $2^{\color{red}{2}^{\color{blue}{2^2}}} = 2^{16} = 65536$.

Answer 2

The difference lies in the applied order of operations.

The Casio calculator does operations strictly left to right unless you break it with parentheses:

2^2^2^2 = 4^2^2 = 16^2 = 256

But Google evaluates the entire expression (correctly) using right-to-left order or operations for the stacked powers:

2^2^2^2 = 2^2^4 = 2^16 = 65536