Irrational Numbers Math Example 5

Follow the full solution, then compare it with the other examples linked below.

Example 5

easy
Is 3+73 + \sqrt{7} rational or irrational?

Solution

  1. 1
    33 is rational, but 7\sqrt{7} is irrational because 7 is not a perfect square.
  2. 2
    A rational number plus an irrational number is irrational, so 3+73 + \sqrt{7} is irrational.

Answer

Irrational\text{Irrational}
Adding a rational number to an irrational number cannot produce a rational result. If it did, subtracting the rational part would incorrectly make the irrational part rational.

About Irrational Numbers

An irrational number is a real number that cannot be expressed as a ratio of two integers pq\frac{p}{q}; its decimal expansion goes on forever without repeating any fixed block of digits.

Learn more about Irrational Numbers →

More Irrational Numbers Examples

Example 1 easy

Classify each number as rational or irrational: [formula], [formula], [formula], [formula].

Example 2 medium

Show that [formula] lies between [formula] and [formula], then estimate it to one decimal place.

Example 3 medium

Prove that [formula] is irrational by contradiction.

Example 4 easy

Is [formula] rational or irrational? Explain.

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