Given $$ x0 $$ and $$ y0 $$, select the expression that is equivalent to $$ \sqrt{16 x^{7} y^{11}} $$

Question

Given $$ x>0 $$ and $$ y>0 $$, select the expression that is equivalent to $$ \sqrt{16 x^{7} y^{11}} $$

UpStudy AI · Accepted Answer

To simplify the expression $$\sqrt{16 x^{7} y^{11}}$$ for $$x > 0$$ and $$y > 0$$, follow these steps:

1. **Break Down the Square Root:**
   $$
   \sqrt{16 x^{7} y^{11}} = \sqrt{16} \cdot \sqrt{x^7} \cdot \sqrt{y^{11}}
   $$

2. **Simplify Each Part:**
   - $$\sqrt{16} = 4$$
   - $$x^7$$ can be written as $$x^6 \cdot x$$, where $$x^6 = (x^3)^2$$. Thus:
     $$
     \sqrt{x^7} = \sqrt{(x^3)^2 \cdot x} = x^3 \sqrt{x}
     $$
   - Similarly, $$y^{11}$$ can be written as $$y^{10} \cdot y$$, where $$y^{10} = (y^5)^2$$. Thus:
     $$
     \sqrt{y^{11}} = \sqrt{(y^5)^2 \cdot y} = y^5 \sqrt{y}
     $$

3. **Combine the Simplified Parts:**
   $$
   4 \cdot x^3 \cdot y^5 \cdot \sqrt{x} \cdot \sqrt{y} = 4 x^3 y^5 \sqrt{xy}
   $$

So, the simplified equivalent expression is:

$$
4 x^{3} y^{5} \sqrt{xy}
$$

**Answer:** 
After simplifying, an equivalent expression is 4 x³ y⁵ multiplied by √( x y ). Thus,

4 x³y⁵ √(xy)
The simplified equivalent expression is $$4x^{3}y^{5}\sqrt{xy}$$.

Given \( x>0 \) and \( y>0 \), select the expression that is equivalent to \[ \sqrt{16 x^{7} y^{11}} \]

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