Question

On a number line, the irrational number x to the power of 2+6
lies between which two whole numbers?
between 12 and 13
between 15 and 16
between 59 and 60
between 83 and 84

Answer 1

To determine the range in which \(x\) can lie such that \(x^2 + 6\) falls between the specified whole numbers, we will solve the inequalities for \(x^2\).

1. **Between 12 and 13**:
\[
12 < x^2 + 6 < 13
\]
Subtracting 6 from all parts gives:
\[
6 < x^2 < 7
\]
Taking the square root:
\[
\sqrt{6} < |x| < \sqrt{7}
\]

2. **Between 15 and 16**:
\[
15 < x^2 + 6 < 16
\]
Subtracting 6:
\[
9 < x^2 < 10
\]
Taking the square root:
\[
3 < |x| < \sqrt{10}
\]

3. **Between 59 and 60**:
\[
59 < x^2 + 6 < 60
\]
Subtracting 6:
\[
53 < x^2 < 54
\]
Taking the square root:
\[
\sqrt{53} < |x| < \sqrt{54}
\]

4. **Between 83 and 84**:
\[
83 < x^2 + 6 < 84
\]
Subtracting 6:
\[
77 < x^2 < 78
\]
Taking the square root:
\[
\sqrt{77} < |x| < \sqrt{78}
\]

Now let's check the approximations of the square roots to determine which intervals correspond to irrational numbers:

- \(\sqrt{6} \approx 2.45\), \(\sqrt{7} \approx 2.65\) → Interval between \(2.45\) and \(2.65\).

- \(\sqrt{9} = 3\), \(\sqrt{10} \approx 3.16\) → Interval between \(3\) and \(3.16\).

- \(\sqrt{53} \approx 7.28\), \(\sqrt{54} \approx 7.35\) → Interval between \(7.28\) and \(7.35\).

- \(\sqrt{77} \approx 8.77\), \(\sqrt{78} \approx 8.83\) → Interval between \(8.77\) and \(8.83\).

### Conclusion
Now we established that:
- For \(x^2 + 6\) being between 12 and 13, \(x\) lies between approximately \(2.45\) and \(2.65\).
- For the other options, similar calculations can be applied.

Thus, the irrational number \(x^2 + 6\) does not fit between 12 and 13 for rational \(x\), meaning approximate values yield the function as predominantly becoming irrational.

However, the expression indicates that \(x^2 + 6\) achieves proximity towards ratio-defined values aligning with specific whole numbers. The arrangement confirms locations.

The intervals imply:
- Between \(83\) and \(84\): The number nearest to a whole number hence fits.

The answer thus confirms:
- The value \(x^2 + 6\) lies best **between 83 and 84**.