Understanding Rational and Irrational Numbers

To determine which of the given options is not an irrational number, we first need to understand what rational and irrational numbers are.

• Rational Number: A number that can be expressed in the form $\frac{p}{q}$, where \(p\) and \(q\) are integers and \(q \neq 0\). Rational numbers have terminating or repeating decimal representations. For example, 3, -5, 1/2, 0.75, 0.333... are rational numbers.
• Irrational Number: A number that cannot be expressed in the form $\frac{p}{q}$. Irrational numbers have non-terminating and non-repeating decimal representations. Examples include $\sqrt{2}$, $\pi$, $\sqrt{7}$. We are looking for the number that is not an irrational number among the options, meaning we are searching for the rational number.

Analyzing Each Option to Identify the Non-Irrational Number

Option 1: $\mathbf{4\sqrt{5}}$

This option is given as $4\sqrt{5}$. The number $\sqrt{5}$ is the square root of a non-perfect square (5). Therefore, $\sqrt{5}$ is an irrational number. When a rational number (which is 4 in this case) is multiplied by an irrational number, the result is always an irrational number (unless the rational number is 0). Thus, $4\sqrt{5}$ is an irrational number.

Option 2: $\mathbf{\sqrt{9} \sqrt{16}}$

This option is given as $\sqrt{9} \sqrt{16}$. Let's evaluate the value of this expression:

• $\sqrt{9}$ is the positive number that, when multiplied by itself, equals 9. The value is 3. The number 3 is an integer and can be written as $\frac{3}{1}$, which fits the definition of a rational number.
• $\sqrt{16}$ is the positive number that, when multiplied by itself, equals 16. The value is 4. The number 4 is an integer and can be written as $\frac{4}{1}$, which is also a rational number.

Now, let's find the product: $\sqrt{9} \sqrt{16} = 3 \times 4 = 12$.

The number 12 can be expressed as $\frac{12}{1}$, where \(p=12\) and \(q=1\). Since 12 can be expressed as a fraction of integers, it is a rational number. Therefore, $\sqrt{9} \sqrt{16}$ is not an irrational number.

Option 3: $\mathbf{\sqrt{11}}$

This option is $\sqrt{11}$. The number 11 is not a perfect square (it cannot be obtained by squaring an integer). Thus, $\sqrt{11}$ is an irrational number.

Option 4: $\mathbf{\sqrt{15}}$

This option is $\sqrt{15}$. The number 15 is not a perfect square. Thus, $\sqrt{15}$ is an irrational number.

Conclusion: Identifying the Number That is Not Irrational

By evaluating each option, we found that $4\sqrt{5}$, $\sqrt{11}$, and $\sqrt{15}$ are all irrational numbers because they involve the square roots of non-perfect squares multiplied by a non-zero rational number. However, the expression $\sqrt{9} \sqrt{16}$ simplifies to $3 \times 4 = 12$, which is a rational number.

The question asks for the option that is not an irrational number. Therefore, the option $\sqrt{9} \sqrt{16}$ is the correct answer as it represents a rational number.