Is the square root of 24 a rational number?

Steps to Solve

1. Identify the definition of a rational number

A rational number is any number that can be expressed as the quotient of two integers, in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

2. Calculate the square root of 24

We need to find $\sqrt{24}$. This can be simplified using the prime factorization of 24:

$$ 24 = 4 \times 6 = 4 \times (2 \times 3) = 2^2 \times 2 \times 3 = 2^3 \times 3 $$

Now, we can simplify $\sqrt{24}$:

$$ \sqrt{24} = \sqrt{2^3 \times 3} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} $$

3. Analyze if the square root of 6 is rational

Next, we need to check if $\sqrt{6}$ can be expressed as a fraction of two integers. The prime factorization of 6 is:

$$ 6 = 2 \times 3 $$

Since both 2 and 3 are primes, $\sqrt{6}$ cannot be simplified into a rational number, which implies that $\sqrt{6}$ is an irrational number.

4. Conclusion about $\sqrt{24}$

Since $\sqrt{24} = 2\sqrt{6}$ and $\sqrt{6}$ is irrational, it follows that $\sqrt{24}$ is also irrational and thus cannot be expressed as a fraction of two integers.