To calculate \( 25^{(4/3)} \), we can break it down as follows:

1. First, rewrite \( 25 \) as \( 5^2 \):
   \[
   25^{(4/3)} = (5^2)^{(4/3)}
   \]

2. Use the power of a power property, \( (a^m)^n = a^{m \cdot n} \):
   \[
   (5^2)^{(4/3)} = 5^{(2 \cdot \frac{4}{3})} = 5^{\frac{8}{3}}
   \]

3. Now, \( 5^{\frac{8}{3}} \) can be expressed as:
   \[
   5^{\frac{8}{3}} = 5^2 \cdot 5^{\frac{2}{3}} = 25 \cdot 5^{\frac{2}{3}}
   \]

4. To find \( 5^{\frac{2}{3}} \), we calculate:
   \[
   5^{\frac{2}{3}} = \sqrt[3]{5^2} = \sqrt[3]{25}
   \]

Thus, \( 25^{(4/3)} \) simplifies to:
\[
25^{(4/3)} = 25 \cdot \sqrt[3]{25}
\]

If you need a numerical approximation, you can calculate:
\[
\sqrt[3]{25} \approx 2.924
\]
So, \( 25 \cdot 2.924 \approx 73.1 \).

Therefore, \( 25^{(4/3)} \approx 73.1 \).

Would you like more details or have any questions? Here are some related questions:

1. What is the significance of fractional exponents in mathematics?
2. How can I simplify other expressions with fractional exponents?
3. What is the process for calculating cube roots?
4. Can you explain the properties of exponents further?
5. How does this apply to real-world problems, such as volume or area? 

**Tip:** When dealing with fractional exponents, remember that the numerator indicates the power, while the denominator indicates the root.

Discover how to calculate 25^(4/3) using properties of exponents and fractional exponents. This detailed explanation includes breaking down the problem step-by-step, leading to an approximate numerical solution of 73.1. Ideal for students in Grades 9-11.

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