$$ \sqrt{25\cdot\sqrt{20\cdot\sqrt{15}}} $$

We are given the expression $ \sqrt{25\cdot\sqrt{20\cdot\sqrt{15}}} $, and we will simplify it step by step.

### Step 1: Work from the innermost square root
Start with the innermost square root:

\[
\sqrt{15} \approx 3.872
\]

### Step 2: Multiply by 20
Now multiply 20 by the value of $ \sqrt{15} $:

\[
20 \cdot \sqrt{15} \approx 20 \cdot 3.872 = 77.44
\]

### Step 3: Take the square root of 77.44
Next, compute the square root of 77.44:

\[
\sqrt{77.44} \approx 8.8
\]

### Step 4: Multiply by 25
Now multiply 25 by the result:

\[
25 \cdot 8.8 = 220
\]

### Step 5: Take the square root of 220
Finally, compute the square root of 220:

\[
\sqrt{220} \approx 14.83
\]

Thus, the simplified value of the expression is approximately $ 14.83 $.

Would you like further clarification or details on any step? Here are some related questions you might consider:

1. What are the exact values of the square roots involved without approximation?
2. How can this expression be simplified if left in radical form?
3. What would change if the initial numbers were different, say 36 instead of 25?
4. How can you approach simplifying nested square roots algebraically?
5. What other methods can be used to evaluate complex square root expressions?

**Tip**: When simplifying complex nested square roots, always work from the inside out!

Simplify the expression: $ \sqrt{25\cdot\sqrt{20\cdot\sqrt{15}}} $

Simplify the nested square root expression $ \sqrt{25 \cdot \sqrt{20 \cdot \sqrt{15}}} $ with a detailed step-by-step approach. This solution covers properties of square roots, multiplication under square roots, and approximation techniques, making it suitable for students in Grades 9-11.

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