Simplify $$ \sqrt{45x4y7} $$ to the form $$ a\sqrt{b} $$


We need to simplify the expression $ \sqrt{45x^4y^7} $ into the form $ a\sqrt{b} $.

### Step 1: Factor inside the square root
Factor $ 45x^4y^7 $ as follows:

\[
45 = 9 \times 5, \quad x^4 \text{ remains the same}, \quad y^7 = y^6 \times y
\]

So the expression becomes:

\[
\sqrt{9 \times 5 \times x^4 \times y^6 \times y}
\]

### Step 2: Separate perfect squares
Now, split the square root into perfect squares and non-perfect squares:

\[
\sqrt{9} \times \sqrt{5} \times \sqrt{x^4} \times \sqrt{y^6} \times \sqrt{y}
\]

### Step 3: Simplify each square root
- $ \sqrt{9} = 3 $
- $ \sqrt{x^4} = x^2 $
- $ \sqrt{y^6} = y^3 $
- $ \sqrt{5} $ and $ \sqrt{y} $ remain as they are since 5 and $ y $ are not perfect squares.

Thus, we have:

\[
3x^2y^3 \sqrt{5y}
\]

### Final Answer:
The simplified expression is:

\[
3x^2y^3 \sqrt{5y}
\]

Let me know if you have any questions or need further details!

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Here are some related questions:
1. How do you simplify square roots with variables?
2. How do you handle fractional exponents?
3. What are the rules for simplifying radicals?
4. Can you explain how to combine like terms under a square root?
5. How would you simplify $ \sqrt{75x^2y^5} $?

**Tip**: Always factor numbers and variables inside square roots to identify perfect squares!

Simplify $ \sqrt{45x^4y^7} $ to the form $ a\sqrt{b} $.

Learn how to simplify the radical expression $ \sqrt{45x^4y^7} $ into the form $ 3x^2y^3 \sqrt{5y} $. This problem involves key concepts like the properties of square roots and the simplification of radicals. Perfect for students in Grades 9-10.

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