Jake runs 76–√

To find the difference in miles that Jake ran each day, we need to calculate the difference between the miles run on Monday and Tuesday.

Jake ran \( \sqrt{76} \) miles on Monday and \( \sqrt{56} \) miles on Tuesday. We want to calculate:

\[
\sqrt{76} - \sqrt{56}
\]

To simplify this expression, we can factor out the square roots:

1. Calculate \( \sqrt{76} \):
\[
\sqrt{76} = \sqrt{4 \times 19} = \sqrt{4} \cdot \sqrt{19} = 2\sqrt{19}
\]

2. Calculate \( \sqrt{56} \):
\[
\sqrt{56} = \sqrt{4 \times 14} = \sqrt{4} \cdot \sqrt{14} = 2\sqrt{14}
\]

Now we have:
\[
\sqrt{76} - \sqrt{56} = 2\sqrt{19} - 2\sqrt{14} = 2(\sqrt{19} - \sqrt{14})
\]

Since none of the response options can be expressed in this simplified radical form, let's check each response to determine which one matches our calculations:

- \( \sqrt{12} \): This square root cannot be broken down easily.
- \( 4\sqrt{12} \): This is likely not simplifying the square roots correctly.
- \( 2\sqrt{12} \): Simplifies to \( 4\sqrt{3} \) and isn't relevant.
- \( 2\sqrt{19} - 2\sqrt{14} \): Directly matches.

Thus, the most appropriate difference based on our calculations is not listed as a direct option, but the correct simplified expression from our calculations should be:

\[
2(\sqrt{19} - \sqrt{14}) \quad \text{(Not in the options)}
\]

However, if the task is simply to select from available choices assuming the simplest forms are close, a close re-evaluation suggests you might need to closely represent logical differences or directly engage their numerical values.

---

If we explicitly calculate the values of \( \sqrt{76} \) and \( \sqrt{56} \):

\[
\sqrt{76} \approx 8.717 \quad and \quad \sqrt{56} \approx 7.483
\]

The numerical difference is approximately \( 8.717 - 7.483 \approx 1.234 \).

This aligns logically nearer mathematical roots where you check through:

Rounding direct solutions may correspond / seem to converge to \( 2(\sqrt{19} - \sqrt{14}) \) as precise but isn't explicitly an option.

Therefore, based on checking values let's say the effective direct choice would potentially be \( 2\sqrt{6} \) or closely checking adjustments against them would likely work best if simplified aligns elsewhere.

Confirm numerical approximations or domain expectations would yield filtering against equal representations or notable entries through if you adjust again since potentially options don't give roots directly back.

Thus, difference calculated stands yet evaluation suggests verifying step-back might yield operative paths.