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what sets of numbers does square root 17 belong to?

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When you take the square root of a number, there are two types of answers you might get: A nice number (a whole number or integer) or a not-nice number (a decimal or fraction)*. If you get a nice whole number, then the square root you started with has been simplified. That number is rational**. If you get a not-so-nice number, then the square root you started with is called irrational.

√(17) is irrational, because it's not a "nice" number. √(64) is rational, because it simplifies to 8, a "nice" number.

Math trivia: In Europe, non-nice-number square roots are called "surds". Yup, really!

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*It's not really true that decimals and fractions are "not nice" but people do tend to think of them that way. Really, they're just numbers like 7 and 12; they just look different - don't discriminate against them. ;)

**Rational numbers are numbers that can be written as ratios. Ratio is really just a fancy word that means fraction.