Let's simplify each expression step-by-step.

Question 1C

The expression is: $(4m^2)(-m^8)(m)$

1. Combine the constants: $4 \times (-1) = -4$.
2. Combine the terms with $m$ by adding exponents:
   • $m^2 \times m^8 \times m^1 = m^{2 + 8 + 1} = m^{11}$.

So, the expression simplifies to: $-4m^{11}$

Question 1D

The expression is: $(2d^5 f)^3 (d^4 f^7)$

1. Start by expanding $(2d^5 f)^3$:

   • The coefficient $2^3 = 8$.
   • For $d$, $(d^5)^3 = d^{15}$.
   • For $f$, $(f)^3 = f^3$.
   • So, $(2d^5 f)^3 = 8d^{15} f^3$.

2. Now multiply $8d^{15} f^3$ by $d^4 f^7$:

   • For $d$, $d^{15} \times d^4 = d^{15 + 4} = d^{19}$.
   • For $f$, $f^3 \times f^7 = f^{3 + 7} = f^{10}$.

Thus, the simplified form is: $8d^{19} f^{10}$

Would you like more details on these steps or have any questions?

Follow-up Questions

1. How do you apply exponent rules to simplify expressions with variables?
2. What is the significance of using the fewest possible exponents in algebra?
3. How would the expressions change if negative exponents were involved?
4. Can you explain the difference between product and power rules in exponents?
5. How does the distributive property work in simplifying expressions like these?

Tip

When simplifying expressions, always handle coefficients and variables separately before combining them back together.