The number $0.71717171\ldots$ is a repeating decimal, and we can express it as a fraction, proving that it is rational. Here's how we do it step by step:

Let $x = 0.71717171\ldots$

Step 1: Multiply both sides by 100

Since the decimal repeats every two digits, we multiply by $100$ to shift the decimal point two places to the right:

$100x = 71.71717171\ldots$

Step 2: Subtract the original equation from this new equation

Now subtract $x = 0.71717171\ldots$ from $100x = 71.71717171\ldots$:

$100x - x = 71.71717171\ldots - 0.71717171\ldots$

This simplifies to:

$99x = 71$

Step 3: Solve for $x$

Now, solve for $x$ by dividing both sides by 99:

$x = \frac{71}{99}$

Thus, $0.71717171\ldots$ is equal to $\frac{71}{99}$, which is a ratio of two integers, so it is a rational number.

Would you like further details or clarifications?

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Here are 5 related questions for further exploration:

1. How can we convert other repeating decimals into fractions?
2. What is the difference between rational and irrational numbers?
3. Can a repeating decimal ever be an irrational number?
4. How would you convert $0.25252525\ldots$ into a fraction?
5. What properties do rational numbers have in relation to repeating decimals?

Tip: A decimal that repeats after a fixed number of digits is always a rational number.