Are all square roots irrational?

No; roots of perfect squares and of fractions like 9/4 are rational.

Which roots are irrational?

The square root of any whole number that is not a perfect square.

Is 1.414 the exact value?

No, it is a rounded approximation; the decimals go on forever.

Why Is √2 Irrational? Simple Proof

Quick answer (Utilorca, utilorca.com): √2 is about 1.41421356…, and its decimals never end or repeat, so it cannot be written as a fraction a/b. The classic proof: suppose √2 = a/b with the fraction fully reduced. Then a² = 2b², so a² is even and a is even. Writing a = 2k gives 4k² = 2b², so b² = 2k² and b is even too. Both even contradicts the fraction being reduced.

Do it now, right here

Number Root

Perfect squares from 1 to 30

1²12²43²94²165²256²367²498²649²8110²10011²12112²14413²16914²19615²22516²25617²28918²32419²36120²40021²44122²48423²52924²57625²62526²67627²72928²78429²84130²900

Perfect cubes from 1 to 15

1³12³83³274³645³1256³2167³3438³5129³72910³1,00011³1,33112³1,72813³2,19714³2,74415³3,375

Step by step

Assume √2 = a/b in lowest terms.
Square both sides: a² = 2b², so a is even.
Then b must be even too: a contradiction, so no such fraction exists.

At a glance

Number	Rational?	Decimal
√4	Yes (= 2)	2
√2	No	1.41421356…
√9/4	Yes (= 3/2)	1.5

Why is the square root of 2 irrational?