Finding the Rational Number Product

The question asks us to identify which product among the given options results in a rational number. A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $p$ is an integer and $q$ is a non-zero integer. Examples include 1, 0, -5, $\frac{1}{2}$, $0.75$. An irrational number, on the other hand, cannot be expressed this way, like $\sqrt{2}$ or $\pi$. We are looking for the option that gives a Rational Number Product.

Analyzing Products of Square Roots

We need to evaluate each product of square roots provided in the options to determine if the result is a rational number or an irrational number. We can use the property of square roots that $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$.

Option 1: $\sqrt{2} \times \sqrt{3}$

Let's calculate the product:

$\sqrt{2} \times \sqrt{3} = \sqrt{2 \times 3} = \sqrt{6}$

The number 6 is not a perfect square (it's not the square of an integer). Therefore, $\sqrt{6}$ is an irrational number. This is not the Rational Number Product we are looking for.

Option 2: $\sqrt{9} \times \sqrt{5}$

Let's calculate the product. We can simplify $\sqrt{9}$ first or multiply directly:

Method 1: Simplifying Radicals first:

$\sqrt{9} \times \sqrt{5} = 3 \times \sqrt{5} = 3\sqrt{5}$

Since $\sqrt{5}$ is an irrational number (5 is not a perfect square), the product $3\sqrt{5}$ is also an irrational number.

Method 2: Multiplying Square Roots first:

$\sqrt{9} \times \sqrt{5} = \sqrt{9 \times 5} = \sqrt{45}$

To see if $\sqrt{45}$ is rational, we can try Simplifying Radicals by finding perfect square factors of 45. $45 = 9 \times 5$.

$\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$

Again, we find the result is $3\sqrt{5}$, which is an irrational number. This is not the desired Rational Number Product.

Option 3: $\sqrt{27} \times \sqrt{3}$

Let's calculate this product of Square Roots:

$\sqrt{27} \times \sqrt{3} = \sqrt{27 \times 3} = \sqrt{81}$

Now, we need to check if 81 is a perfect square. We know that $9 \times 9 = 81$.

$\sqrt{81} = 9$

The number 9 is an integer. An integer can be expressed as a fraction, for example, $9 = \frac{9}{1}$. Therefore, 9 is a rational number. This product results in a Rational Number Product.

Option 4: None of these

Since we found that the product in Option 3 is a rational number, this option is incorrect.

Conclusion on the Rational Number Product

By evaluating each option, we found that the product $\sqrt{27} \times \sqrt{3}$ results in the number 9, which is a rational number. The other products involved an Irrational Number. Thus, the product $\sqrt{27} \times \sqrt{3}$ is the Rational Number Product among the given choices.