To simplify \(\left(64x^{12}\right)^{\frac{2}{3}}\), follow these steps:

1. Apply the power rule: \(\left(a^m\right)^n = a^{m \cdot n}\).

\[
(64x^{12})^{\frac{2}{3}} = 64^{\frac{2}{3}} \cdot \left(x^{12}\right)^{\frac{2}{3}}
\]

2. Simplify \(64^{\frac{2}{3}}\). Since \(64 = 4^3\), we can write:

\[
64^{\frac{2}{3}} = \left(4^3\right)^{\frac{2}{3}} = 4^2 = 16
\]

3. Now simplify \(\left(x^{12}\right)^{\frac{2}{3}}\):

\[
\left(x^{12}\right)^{\frac{2}{3}} = x^{12 \cdot \frac{2}{3}} = x^8
\]

4. Combine the results:

\[
(64x^{12})^{\frac{2}{3}} = 16x^8
\]

Thus, the simplified expression is:

\[
16x^8
\]

Would you like any further details or have any questions?

Here are 5 related questions:
1. How would you simplify \((81x^{9})^{\frac{1}{2}}\)?
2. What is the general rule for simplifying expressions like \((a^m b^n)^p\)?
3. How do fractional exponents relate to roots?
4. Can you express \((125x^6)^{\frac{1}{3}}\) in simplified form?
5. What happens if the exponent in \((64x^{12})^{2/3}\) is negative?

**Tip:** When simplifying expressions with fractional exponents, always remember that \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\).

Discover how to simplify the expression (64x^12)^{2/3} using exponent rules. This problem introduces fractional exponents, powers of integers, and algebraic simplification, making it suitable for students in Grades 9-10. Step-by-step explanation provided.

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