To solve the expression [tex]\(\sqrt[3]{25^3}\)[/tex], let's follow the steps systematically.
### Step 1: Understanding the Expression
The cube root of [tex]\(25^3\)[/tex] can be represented as:
[tex]\[
\sqrt[3]{25^3}
\][/tex]
### Step 2: Simplifying the Expression
We know that taking the cube root of a number raised to the power of 3 essentially 'undoes' the cube, resulting in the original base. Mathematically, the cube root of [tex]\(a^3\)[/tex] is simply [tex]\(a\)[/tex].
Using this property, we simplify [tex]\(\sqrt[3]{25^3}\)[/tex] to:
[tex]\[
\sqrt[3]{25^3} = 25
\][/tex]
### Step 3: Evaluating and Comparing Against Options
With the result [tex]\(25\)[/tex], let's compare this with the given options to identify the correct answer:
1. [tex]\(25^6\)[/tex]
[tex]\[
25^6 = 244140625
\][/tex]
This is far larger than 25, so it is not the correct answer.
2. [tex]\(25 \frac{\frac{\pi}{2}}{2}\)[/tex]
[tex]\[
25 \frac{\frac{\pi}{2}}{2} = 25 \frac{\pi}{4} \approx 25 \frac{3.14159}{4} \approx 25 \times 0.785398 \approx 19.63495
\][/tex]
This is close but not equal to 25, so it is not the correct answer.
3. [tex]\(25^5\)[/tex]
[tex]\[
25^5 = 9765625
\][/tex]
This is far larger than 25, so it is not the correct answer.
4. [tex]\(25^{\frac{2}{3}}\)[/tex]
[tex]\[
25^{\frac{2}{3}} \approx 8.549879733383484
\][/tex]
This is much smaller than 25, so it is not the correct answer.
Given that none of the provided choices equal 25 directly based on standard interpretations or approximations, option 2 might have been intended incorrectly but it seems to be the closest logical interpretation among the given options.
For precision:
- The actual result of [tex]\(\sqrt[3]{25^3} \approx 24.999999999999996\)[/tex], essentially equivalent to 25.
Thus, considering the step-by-step simplification:
- The closest match, though slightly approximate due to numerical intricacies, might indicate an unintended slight deviation or incorrect problem framing on the given options. Strictly speaking based on standards, confirming rechecking the specific lists would ensure the best clarity.
Hence, the directly simplified value remains [tex]\( \boxed{25} \)[/tex], though none match exactly in given multiple choices conventionallydirectly.
