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

Question

UpStudy AI · Accepted Answer

To simplify the expression:

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

let's break it down step by step.

### Step 1: Simplify the Numerator

Start with the numerator:

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

Apply the exponent $$-\frac{3}{2}$$ to each factor inside the parentheses:

$$
16^{-3/2} \cdot \left(x^{2/3}\right)^{-3/2} \cdot \left(y^{-2}\right)^{-3/2}
$$

Calculate each part:

1. **For 16\^{-3/2}:**
   $$
   16^{-3/2} = \left(16^{1/2}\right)^{-3} = 4^{-3} = \frac{1}{64}
   $$

2. **For $$x^{2/3}$$ raised to $$-3/2$$:**
   $$
   \left(x^{2/3}\right)^{-3/2} = x^{(2/3) \cdot (-3/2)} = x^{-1}
   $$

3. **For $$y^{-2}$$ raised to $$-3/2$$:**
   $$
   \left(y^{-2}\right)^{-3/2} = y^{-2 \cdot (-3/2)} = y^{3}
   $$

Putting it all together:

$$
\left(16 x^{2 / 3} y^{-2}\right)^{-3 / 2} = \frac{1}{64} \cdot x^{-1} \cdot y^{3} = \frac{y^{3}}{64x}
$$

### Step 2: Simplify the Entire Expression

Now, substitute the simplified numerator back into the original expression:

$$
\frac{\frac{y^{3}}{64x}}{2 x^{-2} y}
$$

This can be rewritten as:

$$
\frac{y^{3}}{64x} \div (2 x^{-2} y) = \frac{y^{3}}{64x} \cdot \frac{1}{2 x^{-2} y}
$$

Simplify the multiplication:

$$
\frac{y^{3} \cdot 1}{64x \cdot 2 x^{-2} y} = \frac{y^{3}}{128 x \cdot x^{-2} y}
$$

Combine the $$x$$ terms:

$$
x \cdot x^{-2} = x^{-1} = \frac{1}{x}
$$

So, the expression becomes:

$$
\frac{y^{3}}{128 \cdot \frac{1}{x} \cdot y} = \frac{y^{3} \cdot x}{128 y} = \frac{x y^{2}}{128}
$$

### Final Simplified Form

$$
\boxed{\dfrac{x y^{2}}{128}}
$$
$$ \frac{x y^{2}}{128} $$

(a) \( \frac{\left(16 x^{2 / 3} y^{-2}\right)^{-3 / 2}}{2 x^{-2} y} \)