<Question>
Simplify $\frac{(10r^3)(4r^6)}{8r^4}$.
</Question>

<think>
<Plan_1>
Okay, so I need to simplify this algebraic expression: (10r³)(4r⁶) divided by 8r⁴. Let me see. Hmm, alright, let's start by breaking it down step by step. First, I remember that when you multiply terms with exponents, you add the exponents if the base is the same.
So, for the numerator, which is (10r³)(4r⁶), I can multiply the coefficients and the variables separately.

Let me write that out.

<prompt_1.1> Simplify the expression (10r³)(4r⁶) by multiplying the coefficients and adding the exponents of the variables. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
The coefficients are 10 and 4. Multiplying those gives 10 * 4 = 40. Then the variables are r³ and r⁶. Since they have the same base, r, I add the exponents: 3 + 6 = 9. So, the numerator simplifies to 40r⁹.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, the denominator is 8r⁴. So the entire expression becomes 40r⁹ divided by 8r⁴. To simplify this, I can divide the coefficients and subtract the exponents of the like bases.

<prompt_2.1> Divide the coefficients 40 and 8, and subtract the exponents of the variables r⁹ and r⁴. </prompt_2.1>
-----

Wait, let me check if I did everything right. Another way to look at it is to write all the steps out:

<prompt_2.2> Verify the simplification process by writing all steps in detail. </prompt_2.2>

Is there any chance I made a mistake in multiplying the coefficients initially? Let me see.

<prompt_2.3> Check the multiplication of coefficients 10 and 4 and confirm the simplification process. </prompt_2.3>

Alternatively, I could simplify before multiplying. Sometimes, that's easier. Let's see.

<prompt_2.4> Simplify the expression by dividing coefficients and subtracting exponents before multiplying. </prompt_2.4>
</Plan_2>

<Execution_2>
<execution_2.1>
Let's do the coefficients first: 40 divided by 8 is 5. Now for the variables: r⁹ divided by r⁴. Since the bases are the same, I subtract the exponents: 9 - 4 = 5. So, putting that together, the simplified form should be 5r⁵.
</execution_2.1>

<execution_2.2>
Original expression: (10r³ * 4r⁶) / (8r⁴)

Multiply numerator: 10 * 4 = 40, r³ * r⁶ = r^(3+6) = r^9. So numerator is 40r^9.

Denominator: 8r^4.

Now divide: 40r^9 / 8r^4. Divide 40 by 8: 40 ÷ 8 = 5. Subtract exponents: r^(9-4) = r^5. So, 5r^5.
</execution_2.2>

<execution_2.3>
10 times 4 is definitely 40. And r³ times r⁶ is r^(3+6) which is r⁹. So that part is correct. Then dividing by 8r⁴, so 40 divided by 8 is 5, and r⁹ divided by r⁴ is r^(9-4) = r⁵. So 5r⁵.
</execution_2.3>

<execution_2.4>
The original expression is (10r³)(4r⁶)/8r⁴. I can cancel out common factors before multiplying everything out. Let's see, 10 and 8 have a common factor of 2, and 4 and 8 have a common factor of 4. Wait, maybe breaking it down:

First, separate the numerical coefficients from the variables. The numerical part is (10 * 4)/8. The variable part is r³ * r⁶ divided by r⁴. Let's handle the numerical coefficients: 10*4=40, then 40 divided by 8 is 5. Then the variables: r^(3+6-4) = r^5. So again, 5r^5.
</execution_2.4>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Let me double-check to make sure I didn't make any mistakes. Let's verify each part again.

<prompt_3.1> Verify the simplification process by checking each part again. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
First, the numerator: 10r³ times 4r⁶. Coefficients: 10*4=40. Variables: r³*r⁶=r^(3+6)=r⁹. So numerator is 40r⁹. Denominator is 8r⁴. So dividing 40r⁹ by 8r⁴. Coefficients: 40/8=5. Variables: r^(9-4)=r⁵. So yes, 5r⁵.
</execution_3.1>
</Execution_3>

<Final_answer>
Therefore, the simplified form of the given expression is $\boxed{5r^5}$.
</Final_answer>
</think>