(c) $$ \left(\frac{16 x^{9} y^{3}}{x y^{11}}\right)^{-1 / 2} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\left(\frac{16 x^{9} y^{3}}{x y^{11}}\right)^{-1 / 2}
$$

**Step 1: Simplify the Fraction Inside the Parentheses**

$$
\frac{16 x^{9} y^{3}}{x y^{11}} = 16 \frac{x^{9}}{x} \frac{y^{3}}{y^{11}} = 16 x^{8} y^{-8} = \frac{16 x^{8}}{y^{8}}
$$

**Step 2: Apply the Exponent $$-\frac{1}{2}$$ to the Simplified Fraction**

$$
\left(\frac{16 x^{8}}{y^{8}}\right)^{-\frac{1}{2}} = \frac{y^{8}}{16 x^{8}}^{\frac{1}{2}} = \frac{\sqrt{y^{8}}}{\sqrt{16 x^{8}}}
$$

**Step 3: Simplify the Square Roots**

$$
\sqrt{y^{8}} = y^{4} \quad \text{and} \quad \sqrt{16 x^{8}} = 4 x^{4}
$$

**Step 4: Combine the Results**

$$
\frac{y^{4}}{4 x^{4}}
$$

**Final Simplified Expression:**

$$
\boxed{\dfrac{y^{4}}{4\,x^{4}}}
$$
$$ \frac{y^{4}}{4x^{4}} $$

(c) \( \left(\frac{16 x^{9} y^{3}}{x y^{11}}\right)^{-1 / 2} \)