Okay, let's tackle this problem: (4 5/8)^55 multiplied by (8/37)^55. Hmm, first, I need to understand what exactly is being asked here. So, both terms are raised to the 55th power and then multiplied together. Maybe there's a way to simplify this expression before calculating it? Let me think.

First, I recall that when you multiply two expressions with the same exponent, you can combine the bases by multiplying them first and then raising the product to that exponent. So, maybe I can rewrite the expression as (4 5/8 * 8/37)^55. That seems like a good approach. Let me verify that.

Yes, the rule is a^m * b^m = (a*b)^m. So, that should work here. So, combining the two fractions first might simplify things. Let me compute 4 5/8 multiplied by 8/37.

Wait, 4 5/8 is a mixed number. I should convert that to an improper fraction to make the multiplication easier. 4 is equal to 32/8, so 4 5/8 is 32/8 + 5/8 = 37/8. Ah, that's convenient. So, 37/8 multiplied by 8/37. Let me write that out:

(37/8) * (8/37) = ?

Multiplying the numerators: 37 * 8 = 296. Multiplying the denominators: 8 * 37 = 296. So, 296/296 = 1. Wow, that's nice! So, the product inside the parentheses is 1. Therefore, the entire expression becomes 1^55.

And 1 raised to any power is still 1. So, the answer should be 1. Wait, that seems almost too straightforward. Let me double-check each step to make sure I didn't make a mistake.

First step: converting 4 5/8 to an improper fraction. 4 * 8 = 32, plus 5 is 37. So, 37/8. Correct.

Second step: multiplying 37/8 by 8/37. Numerators: 37 * 8 = 296. Denominators: 8 * 37 = 296. So, 296/296 = 1. Correct.

Third step: raising 1 to the 55th power. 1^55 = 1. Correct.

So, all steps check out. Therefore, the final answer is 1. That makes sense because the original two terms are reciprocals of each other when simplified. Let me confirm that.

Wait, 4 5/8 is 37/8, and the other term is 8/37. So, yes, they are reciprocals. When you raise reciprocals to the same exponent and multiply them, you get (a * 1/a)^n = 1^n = 1. Exactly. So, that's why the result is 1. That's a neat property. So, the answer is indeed 1.

I think that's solid. No steps skipped, all checks done. So, the answer should be boxed 1.
Therefore, the value of $\left(4\dfrac{5}{8}\right)^{55} \cdot \left(\dfrac{8}{37}\right)^{55}$ is $\boxed{1}$.
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