<Question>
Simplify: $\frac{\sqrt{2.5^2-0.7^2}}{2.7-2.5}$.
</Question>

<think>
<Plan_1>
Okay, so I need to simplify the expression √(2.5² - 0.7²) divided by (2.7 - 2.5). Let me start by breaking it down step by step. First, I'll look at the numerator, which is the square root of (2.5 squared minus 0.7 squared). Then the denominator is straightforward, just 2.7 minus 2.5.

Let me compute each part separately.

Starting with the denominator: 2.7 minus 2.5.
That's simple, right? 2.7 - 2.5 equals 0.2. So the denominator is 0.2. I can write that as a fraction if needed. Since 0.2 is the same as 1/5, maybe that will help later.

Now the numerator: √(2.5² - 0.7²). Let me calculate inside the square root first. Compute 2.5 squared.

<prompt_1.1> Calculate 2.5 squared and 0.7 squared, then subtract the results. </prompt_1.1>
-----

Wait, but maybe there's another way to approach the numerator? Let me see.

<prompt_1.2> Consider alternative methods to simplify the numerator. </prompt_1.2>

Alternatively, maybe factor the expression under the square root. The numerator is √(2.5² - 0.7²). That resembles the difference of squares, which factors into (2.5 - 0.7)(2.5 + 0.7). Let me try that.

<prompt_1.3> Factor the expression under the square root using the difference of squares. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
2.5 times 2.5. Hmm, 2 times 2 is 4, 0.5 times 0.5 is 0.25, but wait, that's not right. Wait, 2.5 squared is actually (2 + 0.5)², which is 2² + 2*2*0.5 + 0.5² = 4 + 2 + 0.25 = 6.25. So 2.5 squared is 6.25. 

Then 0.7 squared. Let's compute that. 0.7 times 0.7. 0.7*0.7 is 0.49. So 0.7 squared is 0.49. 

Now subtract the two: 6.25 - 0.49. Let me do that subtraction. 6.25 minus 0.49. Well, 6.25 minus 0.40 is 5.85, then minus 0.09 more is 5.76. So the numerator inside the square root is 5.76.
</execution_1.1>

<execution_1.2>
Alternatively, perhaps convert the decimals to fractions. 2.5 is 5/2, and 0.7 is 7/10. Let's try that.

So, (5/2)² - (7/10)². Calculating each term:

(25/4) - (49/100). To subtract these, find a common denominator. The least common denominator of 4 and 100 is 100. So convert 25/4 to 625/100 (since 25*25=625 and 4*25=100). Then subtract 49/100: 625/100 - 49/100 = 576/100. So the square root of 576/100 is √576 / √100 = 24/10 = 12/5 = 2.4.
</execution_1.2>

<execution_1.3>
So, 2.5² - 0.7² = (2.5 - 0.7)(2.5 + 0.7). Let's compute those terms. 2.5 - 0.7 is 1.8, and 2.5 + 0.7 is 3.2. So multiplying those gives 1.8 * 3.2. Let me check if that's the same as before: 1.8 * 3.2. 1 * 3.2 is 3.2, 0.8 * 3.2 is 2.56, so adding them gives 5.76. Yep, same as before. So √5.76 is 2.4.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Then take the square root of 5.76. Hmm, what's the square root of 5.76? Let me think. I know that 2.4 squared is 5.76 because 2.4 times 2.4: 2 times 2 is 4, 2 times 0.4 is 0.8, 0.4 times 2 is another 0.8, and 0.4 times 0.4 is 0.16. Adding those up: 4 + 0.8 + 0.8 + 0.16 = 5.76. So √5.76 is 2.4.
-----

Based on execution_1.3:
Therefore, the numerator is 24/10, which simplifies to 12/5.

<prompt_2.1> Simplify the fraction obtained from the alternative method and compare it with the previous result. </prompt_2.1>

Based on execution_1.1:
So now, the entire expression is 2.4 divided by 0.2. Let me compute that.

<prompt_2.2> Divide 2.4 by 0.2 and verify the result. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
Wait, but 24 divided by 10 is 2.4, which is the same as 12/5. So, 12/5 divided by 1/5 (since 0.2 is 1/5). Dividing fractions: 12/5 divided by 1/5 is the same as 12/5 multiplied by 5/1, which is (12*5)/(5*1) = 60/5 = 12.
</execution_2.1>

<execution_2.2>
2.4 divided by 0.2. Well, 0.2 goes into 2.4 how many times? 0.2 times 10 is 2.0, so 10 times. Then there's 0.4 left. 0.2 goes into 0.4 twice. So total is 10 + 2 = 12. So 2.4 divided by 0.2 is 12. 

Alternatively, 2.4 divided by 0.2 is the same as (2.4 / 0.2) = (24 / 2) = 12. Because if you multiply numerator and denominator by 10, you get 24/2, which is 12.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1, execution_2.2:
So the simplified result is 12. Let me check if there's any possible mistake here. Let me verify each step again.

<prompt_3.1> Verify each step in the calculation to ensure accuracy. </prompt_3.1>

Alternatively, if I use fractions throughout:

Numerator: √(5.76) = 2.4 = 12/5. Denominator: 0.2 = 1/5. So (12/5) divided by (1/5) = 12/5 * 5/1 = 12.
Therefore, the simplified expression is 12.

Is there any alternative way to see this? Maybe by recognizing that √(a² - b²)/(c - d), but in this case, the numbers are specific.
Alternatively, perhaps rationalizing the denominator or something, but in this case, since both numerator and denominator are straightforward, dividing them directly works.

Wait, another thought: maybe the original expression can be simplified algebraically before plugging in numbers? Let me see.

<prompt_3.2> Re-evaluate the original expression and confirm the result through algebraic manipulation. </prompt_3.2>

Let me just check once more with decimal computation:

<prompt_3.3> Perform a final verification of the result using decimal computation. </prompt_3.3>
</Plan_3>

<Execution_3>
<execution_3.1>
First, denominator: 2.7 - 2.5 = 0.2. Correct.

Numerator: 2.5 squared is 6.25, 0.7 squared is 0.49. 6.25 - 0.49 = 5.76. Square root of 5.76 is 2.4. Then 2.4 divided by 0.2 is indeed 12.
</execution_3.1>

<execution_3.2>
The expression is √(2.5² - 0.7²)/(2.7 - 2.5). Let me note that 2.7 - 2.5 is 0.2, which is positive, so there's no issue with the sign. The numerator, as we saw, is 2.4. So 2.4 divided by 0.2 is 12. 

Alternatively, if I write 0.2 as 1/5, then 2.4 divided by (1/5) is 2.4*5 = 12.
</execution_3.2>

<execution_3.3>
Compute 2.5 squared: 6.25. 0.7 squared: 0.49. 6.25 - 0.49 = 5.76. Square root is 2.4. Denominator: 2.7 - 2.5 = 0.2. 2.4 divided by 0.2 is 12. 

Yes, all steps check out. So the simplified form is indeed 12.
</execution_3.3>
</Execution_3>

<Final_answer>
Therefore, the simplified expression is $\boxed{12}$.
</Final_answer>
</think>