To solve the expression \((5a^2b)^6 \times a^3 \times b^4\), we need to apply the laws of exponents and multiplication. Here's a step-by-step solution:

1. **Expand the expression inside the parentheses using the exponent:**

   \[(5a^2b)^6 = 5^6 \times (a^2)^6 \times (b)^6\]

   Applying the exponents:

   \[= 5^6 \times a^{12} \times b^6\]

2. **Multiply the expanded terms with the remaining factors \(a^3\) and \(b^4\):**

   \[5^6 \times a^{12} \times b^6 \times a^3 \times b^4\]

3. **Combine like terms (same bases) by adding the exponents:**

   \[5^6 \times a^{12+3} \times b^{6+4}\]

   \[= 5^6 \times a^{15} \times b^{10}\]

4. **Calculate \(5^6\):**

   \(5^6 = 15625\)

5. **Final expression:**

   \[15625a^{15}b^{10}\]

The final answer is:

\[15625a^{15}b^{10}\]

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If you have any questions or need further clarification, please let me know!

Here are some related questions you might find interesting:

1. How do you apply the power of a product rule in exponents?
2. What is the power of a power rule in exponents?
3. How can you simplify expressions with multiple variables and exponents?
4. What are the steps to multiply polynomials with exponents?
5. How do you handle negative exponents in similar problems?
6. What happens when you divide expressions with the same base and different exponents?
7. How do you deal with coefficients when raising a term to a power?
8. Can you solve an expression with fractional exponents similarly?

**Tip:** When dealing with exponents, always remember to add the exponents for like bases when multiplying and subtract them when dividing.

Learn how to solve the expression (5a²b)⁶ × a³ × b⁴ with a detailed step-by-step solution. This problem covers exponents, multiplication of powers, and the laws of exponents. Suitable for students in Grades 7-9.

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