Explanation
Square Root of 51
Is It Rational or Irrational?
To determine if the square root of 51 is rational or irrational, we need to recall the definition of rational and irrational numbers.
A rational number is any number that can be expressed as the quotient or fraction of two integers, say $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$. On the other hand, an irrational number cannot be expressed as a simple fraction; its decimal form is nonrepeating and nonterminating.
Mathematical Representation
Let's denote the square root of 51 as $\sqrt{51}$.

Suppose, for contradiction, that $\sqrt{51}$ is rational. This means we could write it as:
$\sqrt{51} = \frac{a}{b}$Where:
$a, b \in \mathbb{Z} \quad \text{and} \quad b \neq 0$ 
Squaring both sides, we get:
$51 = \frac{a^2}{b^2} \quad \Rightarrow \quad 51b^2 = a^2$ 
The equation $51b^2 = a^2$ implies that $a^2$ is a multiple of 51.
Prime Factorization
We can factorize 51 as:
$51 = 3 \times 17$From this factorization, any perfect square involving 51 must also have both 3 and 17 as squared factors. However, neither 3 nor 17 is a perfect square.
Conclusion
Since neither 3 nor 17 appears in pairs within $a^2$:
$\sqrt{51} \text{ is not a perfect square, hence it cannot be expressed as } \frac{a}{b}.$Therefore, the square root of 51 is irrational.