To solve \((A^{\frac{2}{3}})^2 \times A^{\frac{1}{4}}\):

1. Simplify \((A^{\frac{2}{3}})^2\):

\[
(A^{\frac{2}{3}})^2 = A^{\frac{2}{3} \times 2} = A^{\frac{4}{3}}
\]

2. Now, multiply \(A^{\frac{4}{3}}\) by \(A^{\frac{1}{4}}\):

\[
A^{\frac{4}{3}} \times A^{\frac{1}{4}} = A^{\left(\frac{4}{3} + \frac{1}{4}\right)}
\]

3. To add the exponents, find a common denominator:

\[
\frac{4}{3} + \frac{1}{4} = \frac{16}{12} + \frac{3}{12} = \frac{19}{12}
\]

So, the simplified form is:

\[
A^{\frac{19}{12}}
\]

The final answer is \(A^{\frac{19}{12}}\).

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. Simplify \((B^{\frac{3}{4}})^2 \times B^{\frac{1}{3}}\).
2. Simplify \((C^{\frac{5}{6}})^3 \times C^{\frac{1}{2}}\).
3. Simplify \((D^{\frac{1}{2}})^4 \times D^{\frac{2}{3}}\).
4. Simplify \((E^{\frac{3}{5}})^2 \times E^{\frac{4}{7}}\).
5. Simplify \((F^{\frac{2}{7}})^3 \times F^{\frac{3}{8}}\).

**Tip:** When multiplying expressions with the same base, add the exponents.

This step-by-step guide simplifies the exponential expression (A⅔)² × A¼ using the power of a power and product of powers rules. Learn how to multiply and add exponents to get the final solution, A^(19/12). Perfect for high school students in Grades 9-11.

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