To simplify the expression $2a^6b^4 \cdot 5a^2b^6$, follow these steps:

Step 1: Multiply the constants

Multiply the coefficients 2 and 5: $2 \times 5 = 10$

Step 2: Apply the product rule for exponents

When multiplying terms with the same base, add the exponents.

• For the base $a$, add the exponents of $a^6$ and $a^2$: $a^6 \cdot a^2 = a^{6+2} = a^8$

• For the base $b$, add the exponents of $b^4$ and $b^6$: $b^4 \cdot b^6 = b^{4+6} = b^{10}$

Final Answer:

$10a^8b^{10}$

Would you like more details on this or have any further questions?

Here are some related questions to explore:

1. How do you simplify expressions with negative exponents?
2. How would the expression change if one of the exponents was negative?
3. What is the general rule for multiplying powers with the same base?
4. How do you simplify expressions involving division of powers?
5. What happens when you multiply terms with different variables but the same exponents?

Tip: When multiplying powers with the same base, always remember to add the exponents.