The expression you provided is:

$\sqrt[3]{^{\:}2^3}$

First, let's break it down:

1. The cube root $\sqrt[3]{x}$ of a number $x$ is the number that, when raised to the power of 3, gives $x$.
2. The exponent $2^3 = 8$, so this becomes: $\sqrt[3]{8}$
3. The cube root of 8 is 2 because $2^3 = 8$.

Thus, the value of $\sqrt[3]{2^3}$ is:

$2$

Let me know if you need further clarification or additional help with this!

Here are some related questions for you:

1. What is $\sqrt[4]{16}$?
2. How would you simplify $\sqrt[3]{8x^6}$?
3. Can you explain the difference between cube roots and square roots?
4. What would happen if the exponent inside the root was a negative number?
5. How do you evaluate the square root of a fraction, like $\sqrt{\frac{4}{9}}$?

Tip: When simplifying roots with exponents, try to rewrite the base as a power that makes the root easier to evaluate!