Let's solve the problem step by step.

The expression given is: $\frac{[(a^2)^3 \times b^3]}{\left( \frac{2b^2}{a} \right)^2}$

Step 1: Simplify the Numerator

The numerator is $(a^2)^3 \times b^3$.

Using the power of a power rule, $(a^m)^n = a^{m \cdot n}$: $(a^2)^3 = a^{2 \cdot 3} = a^6$

So the numerator becomes: $a^6 \times b^3$

Step 2: Simplify the Denominator

The denominator is $\left( \frac{2b^2}{a} \right)^2$.

Applying the power rule to each term inside the parentheses: $\left( \frac{2b^2}{a} \right)^2 = \frac{(2)^2 \cdot (b^2)^2}{(a)^2} = \frac{4b^4}{a^2}$

Step 3: Rewrite the Expression

Now, we have: $\frac{a^6 \times b^3}{\frac{4b^4}{a^2}}$

To simplify a fraction divided by another fraction, we can rewrite it as: $a^6 \times b^3 \times \frac{a^2}{4b^4} = \frac{a^6 \cdot a^2 \cdot b^3}{4b^4}$

Step 4: Combine the Terms

Using the product of powers rule for $a$ and $b$: $a^6 \cdot a^2 = a^{6+2} = a^8$ $b^3 / b^4 = b^{3-4} = b^{-1} = \frac{1}{b}$

So, the expression becomes: $\frac{a^8}{4b}$

Final Answer

$\frac{a^8}{4b}$

Would you like further explanation on any step or have any other questions?

Related Questions

1. How would the result change if the base of $b$ in the numerator were raised to a different power?
2. What if we had an additional term, like $c^5$, in both the numerator and denominator?
3. Can you generalize the rule used in this problem to simplify similar expressions?
4. What if the denominator had been raised to a different power, such as 3?
5. How would you simplify an expression like $\frac{(a^3 \cdot b)^2}{\left( \frac{b^3}{a} \right)^4}$?

Tip

When simplifying expressions with exponents, always apply exponent rules carefully to avoid mistakes, especially when combining terms with different powers.