To calculate the real part \( \text{Re}(68e^{-2.53i}) \), we need to express the complex number in rectangular form. The general form for a complex number in polar form is:

\[
re^{i\theta} = r(\cos\theta + i\sin\theta)
\]

Here, \( r = 68 \) and \( \theta = -2.53 \). Now let's calculate the real part, which is given by \( r\cos(\theta) \).

\[
\text{Re}(68e^{-2.53i}) = 68 \cos(-2.53)
\]

We can calculate \( \cos(-2.53) \), which is equivalent to \( \cos(2.53) \), since cosine is an even function.

\[
\cos(2.53) \approx -0.818
\]

Now multiply this value by 68:

\[
68 \times (-0.818) \approx -55.62
\]

Thus, the real part of \( 68e^{-2.53i} \) is approximately:

\[
\boxed{-55.62}
\]

Would you like more details or have any questions? Here are some related questions for further exploration:

1. What is the imaginary part of \( 68e^{-2.53i} \)?
2. How do you convert polar form to rectangular form in general?
3. How does the sine function affect the imaginary part of a complex number?
4. Can you explain why cosine is an even function?
5. What is the significance of Euler's formula in complex analysis?

**Tip:** The real part of a complex number in polar form is always \( r \cos(\theta) \), where \( r \) is the magnitude and \( \theta \) is the argument (angle).

Learn how to calculate the real part of 68e^{-2.53i} by converting the complex number from polar to rectangular form. This problem involves key concepts in complex numbers, Euler's formula, and trigonometry. Suitable for college-level students or advanced high school learners.

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