Verified Answer :

  • Identify that \(\sqrt{7}\) is an irrational number.
  • Test each option by adding it to \(\sqrt{7}\).
  • Observe that \(\sqrt{7} + (-\sqrt{7}) = 0\), which is a rational number.
  • Conclude that the correct option is \(\boxed{-\sqrt{7}}\).

Explanation

  1. Problem Analysis and Definitions

We are given the irrational number \(\sqrt{7}\) and a set of options: \(0, -\sqrt{7}, 7, \sqrt{7}\). We need to determine which of these options, when added to \(\sqrt{7}\), results in a rational number. Recall that a number is rational if it can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q eq 0\). The square root of any positive integer that is not a perfect square is always an irrational number. Since \(7\) is not a perfect square, \(\sqrt{7}\) is irrational.

  1. Evaluating the Options

We test each option by performing the addition:

  • Option 1: \(0\). The sum is \(\sqrt{7} + 0 = \sqrt{7}\). This is irrational.
  • Option 2: \(-\sqrt{7}\). The sum is \(\sqrt{7} + (-\sqrt{7}) = 0\). We can write \(0 = \frac{0}{1}\), which satisfies the definition of a rational number.
  • Option 3: \(7\). The sum is \(\sqrt{7} + 7\). The sum of an irrational number and a rational number is always irrational.
  • Option 4: \(\sqrt{7}\). The sum is \(\sqrt{7} + \sqrt{7} = 2\sqrt{7}\). Since \(\sqrt{7}\) is irrational, any non-zero rational multiple of it remains irrational.
  1. Conclusion

Based on our evaluation, the only value that results in a rational number when added to \(\sqrt{7}\) is \(-\sqrt{7}\). Thus, the correct option is \(-\sqrt{7}\). \(\boxed{-\sqrt{7}}\).

Examples

Understanding whether a sum is rational or irrational is useful in fields like engineering or physics where you often need to simplify expressions involving radicals. For example, if you are calculating the dimensions of a structure or the frequency of a wave and you arrive at an expression containing \(\sqrt{7}\), knowing what value to add or subtract to "neutralize" that irrational component can help you simplify your final results into a clean, rational number for practical manufacturing or design.

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