4) (xa) Simplify. $$ 2^{-\frac{3}{2}} \cdot 18^{-\frac{3}{2}} $$ $$ \square $$

Question

UpStudy AI · Accepted Answer

To simplify the expression $$ 2^{-\frac{3}{2}} \cdot 18^{-\frac{3}{2}} $$, follow these steps:

1. **Combine the Bases:**
   $$
   2^{-\frac{3}{2}} \cdot 18^{-\frac{3}{2}} = (2 \times 18)^{-\frac{3}{2}} = 36^{-\frac{3}{2}}
   $$

2. **Simplify the Exponent:**
   $$
   36^{-\frac{3}{2}} = \frac{1}{36^{\frac{3}{2}}}
   $$

3. **Evaluate $$ 36^{\frac{3}{2}} $$:**
   $$
   36^{\frac{3}{2}} = \left( \sqrt{36} \right)^3 = 6^3 = 216
   $$

4. **Final Simplification:**
   $$
   \frac{1}{36^{\frac{3}{2}}} = \frac{1}{216}
   $$

**Final Answer:**
$$
\boxed{\dfrac{1}{216}}
$$
$$ \frac{1}{216} $$

4) (xa) Simplify. \( 2^{-\frac{3}{2}} \cdot 18^{-\frac{3}{2}} \) \( \square \)