Just as rational numbers were made up by ratios, the irrational numbers cannot be expressed as a ratio. For example the square root of 2 does not equal one integer divided by another. You can get closer and closer with fractions, but you will never exactly equal √2.
This also means that the decimal expansion of an irrational number goes on forever.
Square roots, cube roots and other roots can all be irrational numbers. Transcendental numbers like pi and e are also irrational.
Raising an integer to a fraction will give an irrational number (unless it comes out exactly and gives an integer.) For example:
21/2 = √2 = 1.41421356...
53/2 = (√5)3 = 11.180339877...
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