Okay, let's see. I need to simplify this fraction: (10r³)(4r⁶) divided by 8r⁴. Hmm, let me start by recalling how to handle exponents and coefficients. 

First, maybe I should multiply the numerators together. The numerator has two terms: 10r³ and 4r⁶. If I multiply those, I can multiply the coefficients and then add the exponents of r since they have the same base. So 10 times 4 is 40. Then for the r terms, r³ times r⁶ is r^(3+6) which is r⁹. So the numerator simplifies to 40r⁹. 

Now the denominator is 8r⁴. So the whole expression becomes 40r⁹ divided by 8r⁴. Let me handle the coefficients first: 40 divided by 8 is 5. Then for the r terms, r⁹ divided by r⁴ is r^(9-4) which is r⁵. So putting it all together, that should be 5r⁵. 

Wait, let me double-check that. If I have (10r³)(4r⁶), multiplying 10 and 4 gives 40, and r³ times r⁶ is r⁹. Then dividing by 8r⁴, so 40 divided by 8 is 5, and r⁹ divided by r⁴ is r⁵. Yeah, that seems right. 

But maybe I can simplify step by step. Let me break it down more. First, the coefficients: 10 times 4 is 40. Then divide by 8: 40 divided by 8 is indeed 5. Then the r terms: numerator is r³ times r⁶. Adding exponents gives 9, and denominator is r⁴, so subtracting exponents 9-4=5. So r⁵. So 5r⁵. 

Alternatively, maybe factor out common terms before multiplying. Let's see. The original expression is (10r³)(4r⁶)/(8r⁴). Let me write it as (10*4*r³*r⁶)/(8*r⁴). Then 10*4 is 40, r³*r⁶ is r⁹, divided by 8r⁴. So same as before. 

Alternatively, maybe simplify the coefficients and the variables separately. Coefficients: 10*4=40, divided by 8=5. Variables: r³*r⁶=r⁹, divided by r⁴=r^(9-4)=r⁵. So 5r⁵. 

I think that's the simplified form. Let me check with plugging in a value for r. Let's say r=2. Original expression: (10*2³)(4*2⁶)/(8*2⁴). Calculate numerator: 10*8=80, 4*64=256. So 80*256=20480. Denominator: 8*16=128. So 20480/128=160. 

Now simplified expression: 5*2⁵=5*32=160. Same result. So that checks out. 

Therefore, the simplified form is 5r⁵. I think that's the answer.
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