Is the square root of 30 a rational number?

Steps to Solve

1. Identify the Nature of the Square Root First, we examine whether $\sqrt{30}$ can be expressed as a fraction. This is true if it’s a rational number, which means it can be represented as $\frac{p}{q}$, where $p$ and $q$ are integers, and $q \neq 0$.

2. Assume it is Rational and Square Both Sides Assume that $\sqrt{30}$ can be expressed as a rational number. If we set $\sqrt{30} = \frac{p}{q}$, squaring both sides gives us: $$ 30 = \frac{p^2}{q^2} $$ This can be rewritten as: $$ p^2 = 30q^2 $$

3. Analyze the Prime Factorization Next, we analyze the prime factorization of 30: $$ 30 = 2 \times 3 \times 5 $$ Since 30 is not a perfect square (its prime factors do not have even powers), $p^2$ must also have the same property, indicating the impossibility of finding integers $p$ and $q$ such that both sides hold true.

4. Conclude on the Nature of the Root Since we have shown that the equation leads to a contradiction involving the prime factors, it implies that $\sqrt{30}$ cannot be expressed as a fraction of two integers, meaning it is an irrational number.