Math Snap

STEP 1

1. The given expression is $4 x^{-3} y^{-1} \sqrt{x y^{5}}$.
2. We need to express this in the form $a x^{b} y^{c}$.
3. We will then find the product of $a$, $b$, and $c$.

STEP 2

1. Simplify the expression under the square root.
2. Simplify the entire expression $4 x^{-3} y^{-1} \sqrt{x y^{5}}$.
3. Identify the values of $a$, $b$, and $c$.
4. Calculate the product of $a$, $b$, and $c$.

STEP 3

Simplify the expression under the square root:
$$\sqrt{x y^{5}} = (x y^{5})^{1/2} = x^{1/2} y^{5/2}$$

STEP 4

Substitute the simplified square root expression back into the original expression:
$$4 x^{-3} y^{-1} \cdot x^{1/2} y^{5/2}$$

STEP 5

Combine the powers of $x$ and $y$:
$$4 \cdot x^{-3 + 1/2} \cdot y^{-1 + 5/2}$$ $$= 4 \cdot x^{-5/2} \cdot y^{3/2}$$

STEP 6

Identify $a$, $b$, and $c$ from the expression $4 x^{-5/2} y^{3/2}$:
$$a = 4, \quad b = -\frac{5}{2}, \quad c = \frac{3}{2}$$

Calculate the product of $a$, $b$, and $c$:
$$a \times b \times c = 4 \times \left(-\frac{5}{2}\right) \times \frac{3}{2}$$ $$= 4 \times -\frac{15}{4}$$ $$= -15$$ The product of $a$, $b$, and $c$ is:
$$\boxed{-15}$$