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The expression you've provided is:

\[
y^3 \sqrt{45x^7y^5}
\]

To simplify this expression, let's break it down step by step:

1. **Factor the square root term**:
   \[
   \sqrt{45x^7y^5}
   \]
   The number 45 can be factored as \(45 = 9 \times 5\), and 9 is a perfect square. The square root of 9 is 3.

   So, the expression inside the square root becomes:
   \[
   \sqrt{9 \times 5 \times x^7 \times y^5} = \sqrt{9} \times \sqrt{5 \times x^7 \times y^5} = 3 \times \sqrt{5x^7y^5}
   \]

2. **Simplify the powers inside the square root**:
   - \(x^7\) can be written as \(x^6 \times x^1\). Since \(x^6\) is a perfect square (\(\sqrt{x^6} = x^3\)), it can be taken out of the square root.
   - Similarly, \(y^5\) can be written as \(y^4 \times y^1\). Since \(y^4\) is a perfect square (\(\sqrt{y^4} = y^2\)), it can also be taken out of the square root.

   So the square root term becomes:
   \[
   3 \times x^3 \times y^2 \times \sqrt{5xy}
   \]

3. **Combine with the outside term**:
   The original expression now becomes:
   \[
   y^3 \times 3 \times x^3 \times y^2 \times \sqrt{5xy}
   \]

   Simplify by combining like terms:
   \[
   3x^3y^{3+2} \sqrt{5xy} = 3x^3y^5\sqrt{5xy}
   \]

Thus, the simplified expression is:

\[
\boxed{3x^3y^5\sqrt{5xy}}
\]

Would you like more details on any of these steps or have any other questions?

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Here are five related questions to explore further:

1. How would you simplify a similar expression if the square root contained a different set of powers?
2. What if the coefficient inside the square root was not factorable into a perfect square?
3. How does the order of operations affect the simplification of radical expressions?
4. Can you think of an example where the variables inside the square root cannot be simplified as easily?
5. What are the general rules for simplifying expressions involving both exponents and radicals?

**Tip:** When simplifying expressions involving radicals and exponents, always look for perfect squares or other perfect powers that can be factored out to simplify the square root.

Simplify the expression y^3 * √(45x^7y^5).

Discover how to simplify the expression y^3 * √(45x^7y^5) with a step-by-step solution. This problem explores algebraic concepts like radicals, exponents, and their properties, suitable for students in Grades 9-11.

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