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What is the simplest form of this fraction?

Let's simplify the given fraction step by step.

The fraction is:

\[ \frac{271^2 - 129^2}{542^2 - 258^2} \]

We can use the difference of squares formula, which states that \(a^2 - b^2 = (a - b)(a + b)\).

Applying this to both the numerator and the denominator:

**Numerator:**
\[ 271^2 - 129^2 = (271 - 129)(271 + 129) \]
\[ = 142 \times 400 \]

**Denominator:**
\[ 542^2 - 258^2 = (542 - 258)(542 + 258) \]
\[ = 284 \times 800 \]

Now, let's rewrite the fraction with these results:

\[ \frac{142 \times 400}{284 \times 800} \]

Next, simplify by canceling out common factors:

First, factor the numerator and the denominator:
\[ 142 \times 400 = 142 \times 4 \times 100 \]
\[ 284 \times 800 = 284 \times 8 \times 100 \]

Now, cancel the common factor of \(100\):
\[ \frac{142 \times 4}{284 \times 8} \]

Next, simplify \(142 \div 284 = 1/2\):
\[ \frac{1 \times 4}{2 \times 8} \]
\[ = \frac{4}{16} \]

Finally, simplify \(4 \div 16 = 1/4\):
\[ = \frac{1}{4} \]

Therefore, the simplest form of the given fraction is:

\[ \boxed{\frac{1}{4}} \]

Would you like any details or have any questions?

Here are 8 related questions:
1. What is the difference of squares formula?
2. How do you simplify a fraction?
3. What are common factors in fractions?
4. What is the process for factoring numbers?
5. Can the difference of squares formula be used for sums?
6. How do you simplify complex fractions?
7. What other algebraic identities are useful for simplifying expressions?
8. What are some applications of the difference of squares in real-world problems?

**Tip:** Always look for common factors to simplify fractions more efficiently.

Learn how to simplify a fraction step-by-step using the difference of squares formula. This problem involves simplifying a complex fraction by applying algebraic identities and canceling out common factors. Suitable for students in Grades 7-9.

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