To determine which radical expression simplifies to \( 2x\sqrt{15y} \), we will simplify each of the given options one by one.
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Option: \(3\sqrt{50x^2y}\)
First, simplify \( \sqrt{50x^2y} \): \[ \sqrt{50x^2y} = \sqrt{25 \cdot 2 \cdot x^2 \cdot y} = \sqrt{25} \cdot \sqrt{2} \cdot \sqrt{x^2} \cdot \sqrt{y} = 5 \cdot \sqrt{2} \cdot x \cdot \sqrt{y} = 5x\sqrt{2y} \] Now multiply by 3: \[ 3\sqrt{50x^2y} = 3 \cdot 5x\sqrt{2y} = 15x\sqrt{2y} \]
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Option: \(\sqrt{60x^2y}\)
Simplifying \( \sqrt{60x^2y} \): \[ \sqrt{60x^2y} = \sqrt{4 \cdot 15 \cdot x^2 \cdot y} = \sqrt{4} \cdot \sqrt{15} \cdot \sqrt{x^2} \cdot \sqrt{y} = 2 \cdot \sqrt{15} \cdot x \cdot \sqrt{y} = 2x\sqrt{15y} \]
This simplifies directly to \(2x\sqrt{15y}\).
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Option: \(5\sqrt{6xy^2}\)
Simplifying \( \sqrt{6xy^2} \): \[ \sqrt{6xy^2} = \sqrt{6} \cdot \sqrt{x} \cdot \sqrt{y^2} = \sqrt{6} \cdot \sqrt{x} \cdot y \] Now multiply by 5: \[ 5\sqrt{6xy^2} = 5y\sqrt{6x} \]
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Option: \(\sqrt{30x^2y}\)
Simplifying \( \sqrt{30x^2y} \): \[ \sqrt{30x^2y} = \sqrt{30} \cdot \sqrt{x^2} \cdot \sqrt{y} = \sqrt{30} \cdot x \cdot \sqrt{y} \]
Now comparing all the simplified forms:
- \( 15x\sqrt{2y} \) from option 1,
- \( 2x\sqrt{15y} \) from option 2,
- \( 5y\sqrt{6x} \) from option 3,
- \( \sqrt{30}x\sqrt{y} \) from option 4.
Only option 2: \( \sqrt{60x^2y} \) simplifies to \( 2x\sqrt{15y} \).
Thus, the correct option is: \(\sqrt{60x^2y}\).
3 answers