Verified Answer :

  • A rational number is a ratio of two integers, while \(\sqrt{7}\) is irrational.
  • Adding \(-\sqrt{7}\) to \(\sqrt{7}\) cancels the irrational part: \(\sqrt{7} + (-\sqrt{7}) = 0\).
  • Since \(0\) can be written as \(\frac{0}{1}\), it is a rational number.
  • The correct option is \(\boxed{-\sqrt{7}}\).

Explanation

  1. Analysis of Irrational and Rational Numbers

The objective is to identify which value, when added to \(\sqrt{7}\), results in a rational number. A rational number is defined as a number that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q eq 0\). The number \(\sqrt{7}\) is an irrational number because its decimal representation is non-terminating and non-repeating, and it cannot be expressed as a ratio of two integers. To make an irrational number rational, we must add another number that cancels out the irrational component.

  1. Testing the Options

We test each of the given options: \(0, -\sqrt{7}, 7, \sqrt{7}\).

  1. For \(0\): \(\sqrt{7} + 0 = \sqrt{7}\). Since \(\sqrt{7}\) is irrational, this result is irrational.
  2. For \(-\sqrt{7}\): \(\sqrt{7} + (-\sqrt{7}) = \sqrt{7} - \sqrt{7} = 0\). We can express \(0\) as \(\frac{0}{1}\), which is a rational number.
  3. For \(7\): \(\sqrt{7} + 7\) is the sum of an irrational and a rational number, which is always irrational.
  4. For \(\sqrt{7}\): \(\sqrt{7} + \sqrt{7} = 2\sqrt{7}\). Since \(\sqrt{7}\) is irrational, \(2\sqrt{7}\) is also irrational.
  1. Final Conclusion

After evaluating all the provided options, we found that adding \(-\sqrt{7}\) to \(\sqrt{7}\) yields \(0\), which is a rational number. Therefore, the correct choice is \(-\sqrt{7}\).

Examples

In engineering, when dealing with complex calculations involving square roots like \(\sqrt{7}\) representing physical measurements (e.g., side lengths in non-rational units), adding the negative of that value to "clear" the irrationality allows you to reset a coordinate system to zero. This is a practical application of additive inverses in algebraic manipulation of physical dimensions.