Question

Determine if square root of, 21, end square root
21
​
is rational or irrational and give a reason for your answer.
Answer
Attempt 1 out of 2

The number square root of, 21, end square root
21
​
is
rational
because
it is the square root of a perfect square

Answer 1

The number \( \sqrt{21} \) is actually **irrational**. A rational number can be expressed as the ratio of two integers (a fraction), while an irrational number cannot be expressed in this way.

The reason \( \sqrt{21} \) is considered irrational is that 21 is not a perfect square. The square roots of perfect squares (such as 1, 4, 9, 16, etc.) result in whole numbers, while the square root of 21 does not simplify to a whole number or a simple fraction.

To confirm, there are no two integers \( a \) and \( b \) such that \( \left(\frac{a}{b}\right)^2 = 21 \). Since 21 is not a perfect square, \( \sqrt{21} \) is irrational.