Let $$ f(x)=2 x^{3} $$ and $$ g(x)=\sqrt[3]{x} $$. Find $$ (f g)(x) $$ and $$ \left(\frac{f}{g}\right)(x) $$. Then evaluate $$ f g $$ and $$ \frac{f}{g} $$ for $$ x=-27 $$. $$ (f g)(x)=\square ;(f g)(-27)=118098 $$ $$ \left(\frac{f}{g}\right)(x)=\square ;\left(\frac{f}{g}\right)(-27)=13122 $$ State the domain of $$ (f g)(x) $$ and $$ \left(\frac{f}{g}\right)(x) $$.

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To solve the problem, we will first find the expressions for $$ (f g)(x) $$ and $$ \left(\frac{f}{g}\right)(x) $$.

1. **Finding $$ (f g)(x) $$**:
   $$
   (f g)(x) = f(x) \cdot g(x) = (2x^3) \cdot \left(\sqrt[3]{x}\right)
   $$
   We can rewrite $$ g(x) $$ as $$ x^{1/3} $$:
   $$
   (f g)(x) = 2x^3 \cdot x^{1/3} = 2x^{3 + 1/3} = 2x^{10/3}
   $$

2. **Finding $$ \left(\frac{f}{g}\right)(x) $$**:
   $$
   \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{2x^3}{\sqrt[3]{x}} = \frac{2x^3}{x^{1/3}} = 2x^{3 - 1/3} = 2x^{8/3}
   $$

Now we have:
$$
(f g)(x) = 2x^{10/3}
$$
$$
\left(\frac{f}{g}\right)(x) = 2x^{8/3}
$$

Next, we will evaluate these functions at $$ x = -27 $$.

3. **Evaluating $$ (f g)(-27) $$**:
   $$
   (f g)(-27) = 2(-27)^{10/3}
   $$
   First, we calculate $$ (-27)^{10/3} $$:
   $$
   (-27)^{10/3} = \left((-27)^{1/3}\right)^{10} = (-3)^{10} = 59049
   $$
   Therefore,
   $$
   (f g)(-27) = 2 \cdot 59049 = 118098
   $$

4. **Evaluating $$ \left(\frac{f}{g}\right)(-27) $$**:
   $$
   \left(\frac{f}{g}\right)(-27) = 2(-27)^{8/3}
   $$
   First, we calculate $$ (-27)^{8/3} $$:
   $$
   (-27)^{8/3} = \left((-27)^{1/3}\right)^{8} = (-3)^{8} = 6561
   $$
   Therefore,
   $$
   \left(\frac{f}{g}\right)(-27) = 2 \cdot 6561 = 13122
   $$

Now we summarize the results:
$$
(f g)(x) = 2x^{10/3}; \quad (f g)(-27) = 118098
$$
$$
\left(\frac{f}{g}\right)(x) = 2x^{8/3}; \quad \left(\frac{f}{g}\right)(-27) = 13122
$$

5. **Stating the domain**:
- The domain of $$ (f g)(x) = 2x^{10/3} $$ is all real numbers $$ x $$ since there are no restrictions on $$ x $$ for this expression.
- The domain of $$ \left(\frac{f}{g}\right)(x) = 2x^{8/3} $$ is all real numbers $$ x $$ except $$ x = 0 $$ because $$ g(x) = \sqrt[3]{x} $$ cannot be zero (it would lead to division by zero).

Thus, the final answers are:
$$
(f g)(x) = 2x^{10/3}; \quad (f g)(-27) = 118098
$$
$$
\left(\frac{f}{g}\right)(x) = 2x^{8/3}; \quad \left(\frac{f}{g}\right)(-27) = 13122
$$
Domain of $$ (f g)(x) $$: all real numbers; Domain of $$ \left(\frac{f}{g}\right)(x) $$: all real numbers except $$ x = 0 $$.
$$ (f g)(x) = 2x^{10/3}; \quad (f g)(-27) = 118098 $$ $$ \left(\frac{f}{g}\right)(x) = 2x^{8/3}; \quad \left(\frac{f}{g}\right)(-27) = 13122 $$ Domain of $$ (f g)(x) $$: all real numbers; Domain of $$ \left(\frac{f}{g}\right)(x) $$: all real numbers except $$ x = 0 $$.

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