Okay, let's tackle this math problem step by step. The question is to express \(5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1\) as an integer. Hmm, exponents and operations here. Let me recall the order of operations—PEMDAS or BODMAS. That's Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). So, in this expression, we have exponents, division, multiplication, subtraction, and addition. Let's break it down.

First, let's simplify each term individually. Starting with \(5^5 \div 5^4\). When you divide exponents with the same base, you subtract the exponents. So, \(5^{5-4} = 5^1 = 5\). Okay, that simplifies nicely.

Next term is \(5^3\). That's straightforward—\(5 \times 5 \times 5 = 125\).

Then we have \(5^2 \cdot 5^1\). Again, when multiplying exponents with the same base, you add the exponents. So, \(5^{2+1} = 5^3 = 125\). Wait, that's the same as the second term. Interesting.

So substituting all back into the original expression, we have:

\(5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1 = 5 - 125 + 125\).

Now, let's compute this step by step. Starting from left to right:

First operation: 5 minus 125. That gives -120. Then, add 125 to that result: -120 + 125 = 5.

Wait, so the final result is 5? Let me verify that again to make sure I didn't make a mistake.

Original expression:

\(5^5 \div 5^4 = 5^{5-4} = 5^1 = 5\).

Then subtract \(5^3 = 125\): 5 - 125 = -120.

Then add \(5^2 \cdot 5^1 = 5^{3} = 125\): -120 + 125 = 5.

Yep, that seems right. So the entire expression simplifies to 5. Let me check if there's another way to approach this to cross-verify.

Alternatively, maybe expand each term first:

\(5^5 = 3125\), \(5^4 = 625\), \(5^3 = 125\), \(5^2 = 25\), \(5^1 = 5\).

So substituting:

3125 ÷ 625 - 125 + 25 × 5.

Calculating each operation:

3125 ÷ 625 = 5.

25 × 5 = 125.

So now expression is 5 - 125 + 125. Which is the same as before. 5 - 125 is -120, plus 125 is 5. Yep, same result.

Alternatively, maybe use properties of exponents differently. Let's see, the original expression:

\(5^5 \div 5^4 - 5^3 + 5^2 \cdot 5^1\).

Simplify division first: 5^{5-4} = 5^1 = 5.

Then the multiplication: 5^2 * 5^1 = 5^{3} = 125.

So expression becomes 5 - 125 + 125. Which again is 5.

Another way: Maybe factor out common terms. Let's see. The expression is 5 - 125 + 125. The -125 and +125 cancel each other out, leaving 5. So that's even simpler. So regardless of how you approach it, the result is 5.

I think that's solid. Let me just check once more with actual numbers:

5^5 is 3125, divided by 5^4 (625) is 5. Then subtract 5^3 (125) gives 5 - 125 = -120. Then add 5^2 * 5^1, which is 25*5=125. So -120 + 125 = 5. Yep, same result.

So all methods converge to 5. Therefore, the answer should be 5.
Therefore, the final answer is $\boxed{5}$.
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