To simplify the expression:

$2^{-\frac{1}{3}} \cdot 2^{\frac{7}{3}}$

we apply the rule for multiplying powers with the same base, which states:

$a^m \cdot a^n = a^{m+n}$

Here, the base is 2, and the exponents are $-\frac{1}{3}$ and $\frac{7}{3}$. Adding these exponents:

$-\frac{1}{3} + \frac{7}{3} = \frac{-1 + 7}{3} = \frac{6}{3} = 2$

Thus, the simplified expression becomes:

$2^2 = 4$

So, the simplified form of the expression is:

$4$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you simplify powers with the same base but different exponents?
2. What happens when the exponents are negative in a similar problem?
3. How would you approach the problem if the base was different?
4. How do you handle the subtraction of fractions with different denominators in exponent rules?
5. What happens if you have a division instead of multiplication for powers?

Tip: Always remember that when multiplying powers with the same base, simply add the exponents.