Let's simplify the given expression step by step:

Expression:

$\left( (a)^{-1} \times (b)^{1/3} \times (a)^{-4/3} \times (b)^2 \right)^2$

Step 1: Simplify the terms inside the parentheses

Group the terms with the same base: $a^{-1} \times a^{-4/3} \quad \text{and} \quad b^{1/3} \times b^2$

Now, apply the rule for multiplying powers of the same base $(x^m \times x^n = x^{m+n})$:

For $a$: $a^{-1} \times a^{-4/3} = a^{-1 - 4/3} = a^{-7/3}$

For $b$: $b^{1/3} \times b^2 = b^{1/3 + 2} = b^{1/3 + 6/3} = b^{7/3}$

Thus, the expression inside the parentheses becomes: $a^{-7/3} \times b^{7/3}$

Step 2: Apply the outer exponent

Now raise the entire expression to the power of 2: $\left( a^{-7/3} \times b^{7/3} \right)^2$

Apply the power rule $(x^m)^n = x^{m \times n}$ to both terms: $a^{-7/3 \times 2} \times b^{7/3 \times 2} = a^{-14/3} \times b^{14/3}$

Final Simplified Expression:

$a^{-14/3} \times b^{14/3}$

This is the simplified form of the given expression.

Would you like any further details or explanations?

Here are 5 related questions to deepen your understanding:

1. How do you simplify expressions with negative exponents?
2. What happens when you multiply exponents with the same base?
3. How do fractional exponents relate to radicals?
4. Can you explain why the power rule $(x^m)^n = x^{mn}$ works?
5. How can expressions involving both fractions and negative exponents be written as radicals?

Tip: Always combine terms with the same base before applying any exponents or simplifications.