Let's solve the given expression step by step.

The expression to simplify is:

$\frac{38(b^4)^2}{2c^{-5}} \times \frac{3b^7c^6}{c^2}$

Step 1: Simplify each term

1. $(b^4)^2 = b^{4 \times 2} = b^8$, so the first fraction becomes: $\frac{38b^8}{2c^{-5}}$

2. The second fraction simplifies as: $\frac{3b^7c^6}{c^2} = 3b^7c^{6-2} = 3b^7c^4$

Thus, the expression becomes: $\left(\frac{38b^8}{2c^{-5}}\right) \times \left(\frac{3b^7c^4}{1}\right)$

Step 2: Combine coefficients

For the coefficients: $\frac{38}{2} \times 3 = 19 \times 3 = 57$

For $b$-terms: $b^8 \times b^7 = b^{8+7} = b^{15}$

For $c$-terms: $c^{-5} \times c^4 = c^{-5+4} = c^{-1}$

Step 3: Combine all simplified terms

The result is: $57b^{15}c^{-1}$

Since $c^{-1}$ is equivalent to $\frac{1}{c}$, the final simplified expression is: $\frac{57b^{15}}{c}$

Step 4: Match with the answer choices

The correct answer is: $\boxed{\text{B) } \frac{57b^{15}}{c}}$

Would you like further clarification or additional examples?

Related Questions:

1. What are the properties of exponents used in this problem?
2. How do you simplify fractions with negative exponents?
3. Why is $c^{-1}$ written as $\frac{1}{c}$?
4. Can you explain how multiplication of coefficients works in fractions?
5. What happens if the bases in exponents are different?

Tip:

Always simplify each component of the expression separately (coefficients, variables, and exponents) before combining the terms.