To simplify \( \sqrt{20y^7} \), we can break it down into its components:
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Simplify the number under the square root:
- \( 20 \) can be factored as \( 4 \times 5 \).
- Since \( 4 \) is a perfect square, we can write: \[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \]
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Simplify the variable:
- For \( y^7 \), we can express it as \( (y^6) \times y = (y^3)^2 \times y \).
- Therefore, we can simplify it as follows: \[ \sqrt{y^7} = \sqrt{(y^3)^2 \times y} = \sqrt{(y^3)^2} \times \sqrt{y} = y^3 \sqrt{y} \]
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Combine the results: Putting the simplified components together, we have: \[ \sqrt{20y^7} = \sqrt{20} \cdot \sqrt{y^7} = 2\sqrt{5} \cdot (y^3 \sqrt{y}) = 2y^3\sqrt{5y} \]
So, the simplified form of \( \sqrt{20y^7} \) is: \[ \boxed{2y^3\sqrt{5y}} \]