Okay, let's see. I need to find the eighth term of the given geometric sequence. The sequence provided is 125/9, 25/3, 5, 3, and so on. Hmm, geometric sequence, so each term is multiplied by a common ratio to get the next term. First, I should confirm the common ratio. Let me check between the first two terms. The first term is 125/9, and the second term is 25/3. To find the ratio, I can divide the second term by the first term. So, (25/3) divided by (125/9). Dividing fractions is the same as multiplying by the reciprocal, so that's (25/3) * (9/125). Let's compute that. 25 and 125 can be simplified, 25 divided by 25 is 1, 125 divided by 25 is 5. Similarly, 9 and 3 can be simplified, 9 divided by 3 is 3, 3 divided by 3 is 1. So, the calculation becomes (1/1) * (3/5) = 3/5. So the common ratio r is 3/5. Let me check with the next terms to make sure. The third term is 5. If I multiply the second term 25/3 by 3/5, that should give 5. Let's compute: (25/3) * (3/5) = (25*3)/(3*5) = 75/15 = 5. Correct. Then the fourth term is 3. Multiplying the third term 5 by 3/5 gives 3. Yes, that works. So the common ratio is indeed 3/5. 

Now, the general formula for the nth term of a geometric sequence is a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number. Here, a_1 is 125/9, r is 3/5, and we need the eighth term, so n=8. Plugging into the formula: a_8 = (125/9) * (3/5)^(8-1) = (125/9) * (3/5)^7. 

Let me compute (3/5)^7 first. That's 3^7 divided by 5^7. Let's calculate 3^7. 3^1=3, 3^2=9, 3^3=27, 3^4=81, 3^5=243, 3^6=729, 3^7=2187. Similarly, 5^7. 5^1=5, 5^2=25, 5^3=125, 5^4=625, 5^5=3125, 5^6=15625, 5^7=78125. So (3/5)^7 is 2187/78125. 

Now multiply that by 125/9. So (125/9) * (2187/78125). Let's simplify this. Let's see if there are common factors we can cancel out. 125 and 78125: 78125 divided by 125. Let's compute 125*600=75000, 78125-75000=3125. Then 125*25=3125. So total is 600+25=625. So 125*625=78125. Therefore, 78125 divided by 125 is 625. So 125/78125 is 1/625. Similarly, 2187 and 9: 2187 divided by 9. Let's compute 9*243=2187. So 2187/9=243. Therefore, the multiplication becomes (1/9 * 125) * (2187/78125) = (125/9) * (2187/78125) = (125*2187)/(9*78125). But since 125/78125 = 1/625, and 2187/9 = 243, we can rewrite this as (1/625) * 243 = 243/625. Let me verify that. 125 * 2187 = 125*2000 + 125*187 = 250000 + 23375 = 273375. Then 9*78125 = 703125. So 273375 / 703125. Let's divide numerator and denominator by 273375. 273375 ÷ 273375 = 1, 703125 ÷ 273375. Let's see, 273375*2 = 546750, 703125 - 546750 = 156375. 273375 ÷ 156375 = 1.75. Wait, this seems messy. Maybe my initial approach was better. 

Alternatively, let's break down the fractions:

(125/9) * (2187/78125) = (125/78125) * (2187/9) = (1/625) * (243) = 243/625. Yes, that's correct. Because 125 divided by 78125 is 1/625, and 2187 divided by 9 is 243. So multiplying those gives 243/625. Let me check 243/625. 243 is 3^5, and 625 is 5^4. So that's the simplified fraction. 

Therefore, the eighth term is 243/625. Let me just confirm once more. The first term is 125/9, ratio 3/5. So terms go 125/9, 25/3, 5, 3, 15/5=3? Wait, no. Wait the fourth term is 3. Then fifth term would be 3*(3/5)=9/5, sixth term 27/25, seventh term 81/125, eighth term 243/625. Yes, that's correct. So starting from term 1: 125/9, term 2: 25/3, term3:5, term4:3, term5:9/5, term6:27/25, term7:81/125, term8:243/625. Yes, that's the eighth term. So the answer should be 243/625.

**Final Answer**
The eighth term of the sequence is \boxed{\dfrac{243}{625}}.
Therefore, the eighth term of the sequence is $\boxed{\frac{243}{625}}$.
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