The expression provided is:

$\left( 64 m^3 n^{-6} \right)^{1/3} \cdot \left( m^{-1/3} n^{5/6} \right)^6$

Let's simplify this step-by-step.

Step 1: Simplifying the first part $\left( 64 m^3 n^{-6} \right)^{1/3}$

• The cube root of 64 is $\sqrt[3]{64} = 4$.
• Apply the cube root to each exponent individually: $\left( 64 m^3 n^{-6} \right)^{1/3} = 4 m^{3/3} n^{-6/3} = 4 m n^{-2}$

Step 2: Simplifying the second part $\left( m^{-1/3} n^{5/6} \right)^6$

• Apply the exponent 6 to both terms inside the parentheses: $\left( m^{-1/3} n^{5/6} \right)^6 = m^{-1/3 \cdot 6} n^{5/6 \cdot 6} = m^{-2} n^5$

Step 3: Combine the two parts

Now we multiply the simplified results: $(4 m n^{-2}) \cdot (m^{-2} n^5)$

• Combine the coefficients: $4$
• Combine the powers of $m$: $m^{1} \cdot m^{-2} = m^{1 + (-2)} = m^{-1}$
• Combine the powers of $n$: $n^{-2} \cdot n^5 = n^{-2 + 5} = n^3$

So, the simplified expression is: $4 m^{-1} n^3$

Or equivalently: $\frac{4 n^3}{m}$

Would you like further details on any part of the simplification?

Follow-up Questions:

1. How do you handle negative exponents in a fraction?
2. Can you explain the rules for combining powers of the same base?
3. What is the cube root of other common numbers like 27 or 8?
4. How does distributing exponents work across products?
5. Could you apply this process to similar expressions involving fractional exponents?

Tip: Always simplify inside parentheses first before applying outside exponents when dealing with nested exponents!