The expression in the image can be written as:

$\frac{2^{\frac{1}{2}} \cdot 3^{-\frac{2}{3}} \cdot 6^{-1}}{2^{7} \cdot 3^{\frac{2}{3}} \cdot 6^{-3}}$

We will simplify the expression step by step by applying exponent rules and handling the terms involving base 2, base 3, and base 6.

Step-by-Step Solution:

1. Simplify the numerator and denominator: The terms in the numerator are: $2^{\frac{1}{2}}, 3^{-\frac{2}{3}}, 6^{-1}$ The terms in the denominator are: $2^{7}, 3^{\frac{2}{3}}, 6^{-3}$

2. Break 6 into prime factors: Since $6 = 2 \times 3$, we can rewrite the powers of 6: $6^{-1} = 2^{-1} \times 3^{-1}, \quad 6^{-3} = 2^{-3} \times 3^{-3}$

3. Substitute and combine terms: Now substitute the factorization of 6 into the expression: $\frac{2^{\frac{1}{2}} \cdot 3^{-\frac{2}{3}} \cdot 2^{-1} \cdot 3^{-1}}{2^{7} \cdot 3^{\frac{2}{3}} \cdot 2^{-3} \cdot 3^{-3}}$

4. Simplify powers of 2: Combine all terms involving powers of 2: $2^{\frac{1}{2} - 1 - 7 + 3} = 2^{-\\frac{11}{2}}$

5. Simplify powers of 3: Combine all terms involving powers of 3: $3^{-\frac{2}{3} - 1 - \frac{2}{3} + 3} = 3^{\frac{5}{3}}$

6. Final expression: The simplified expression is: $2^{-\frac{11}{2}} \cdot 3^{\frac{5}{3}}$

Final Answer:

$\boxed{\frac{3^{\frac{5}{3}}}{2^{\frac{11}{2}}}}$

Would you like a more detailed breakdown or have any further questions?

Related Questions:

1. How do you simplify expressions with fractional exponents?
2. What is the rule for dividing exponents with the same base?
3. How do you factor powers of composite numbers?
4. Can you explain how to handle negative exponents?
5. How do you convert a radical to a fractional exponent?

Tip: Always break composite numbers like 6, 12, etc., into their prime factors when simplifying expressions involving exponents.