Simplify the given square root. $$ \sqrt{75 x^{11} y^{6}} $$

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UpStudy AI · Accepted Answer

To simplify the square root $$ \sqrt{75 x^{11} y^{6}} $$, follow these steps:

1. **Factor the Number and Variables:**
   - **75:** $$ 75 = 25 \times 3 = 5^2 \times 3 $$
   - **$$ x^{11} $$:** $$ x^{11} = x^{10} \times x = (x^5)^2 \times x $$
   - **$$ y^{6} $$:** $$ y^{6} = (y^3)^2 $$

2. **Apply the Square Root to Each Factor:**
   - $$ \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} $$
   - $$ \sqrt{x^{11}} = \sqrt{(x^5)^2 \times x} = x^5\sqrt{x} $$
   - $$ \sqrt{y^{6}} = y^3 $$

3. **Combine the Simplified Parts:**
   $$
   \sqrt{75 x^{11} y^{6}} = 5\sqrt{3} \times x^5\sqrt{x} \times y^3 = 5x^5 y^3 \sqrt{3x}
   $$

**Final Simplified Form:**
$$
5x^{5}y^{3}\sqrt{3x}
$$
$$ 5x^{5}y^{3}\sqrt{3x} $$

Simplify the given square root.

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