Q9(ii):

Classify the following numbers as rational or irrational : (ii) $\sqrt{225}$

Initial Setup & Mathematical Definitions

We are tasked with classifying the real number $\sqrt{225}$ as either a rational or an irrational number. To do this rigorously, we must evaluate the radical expression and apply the formal definitions of the real number system.

• Rational Number: A number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers ($p, q \in \mathbb{Z}$) and $q \neq 0$.
• Irrational Number: A real number that cannot be expressed as a simple fraction of two integers. Its decimal expansion is non-terminating and non-repeating.

Step 1: Prime Factorization of the Radicand

The number inside the square root symbol is called the radicand. Here, the radicand is $225$. To simplify the square root, we first determine the prime factorization of $225$ [Per the Fundamental Theorem of Arithmetic].

We systematically divide $225$ by the smallest possible prime numbers:

• $225$ ends in $5$, but let us check divisibility by $3$. The sum of the digits is $2 + 2 + 5 = 9$, which is divisible by $3$.
• $225 \div 3 = 75$
• $75 \div 3 = 25$
• $25 \div 5 = 5$
• $5 \div 5 = 1$

Thus, the prime factorization of $225$ is:

$225 = 3 \times 3 \times 5 \times 5 = 3^2 \times 5^2$

Step 2: Simplification of the Radical Expression

Substitute the prime factorization back into the radical expression. Using the property of radicals $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ (where $a, b \ge 0$):

$\sqrt{225} = \sqrt{3^2 \times 5^2}$

$\sqrt{225} = \sqrt{3^2} \times \sqrt{5^2}$

Since the square root of a squared positive real number is the number itself ($\sqrt{x^2} = x$ for $x \ge 0$):

$\sqrt{225} = 3 \times 5 = 15$

Step 3: Geometric Interpretation of a Perfect Square

Geometrically, finding the square root of $225$ is equivalent to finding the side length of a square whose total area is $225$ square units. Because the side length evaluates to exactly $15$ (a whole number), $225$ is classified as a perfect square.

Step 4: Classification Based on Number Theory

We have established that $\sqrt{225} = 15$. We must now determine if $15$ fits the definition of a rational number.

Any integer $n$ can be written as a fraction by placing it over $1$ ($n = \frac{n}{1}$). Therefore:

$15 = \frac{15}{1}$

In this fraction:

• $p = 15$, which is an integer ($15 \in \mathbb{Z}$).
• $q = 1$, which is an integer ($1 \in \mathbb{Z}$).
• The denominator $q \neq 0$ ($1 \neq 0$).

Because $\sqrt{225}$ can be expressed exactly as the ratio of two integers, it perfectly satisfies the criteria for being a rational number.

Final Solution: $\sqrt{225}$ simplifies to $15$, which can be written as $\frac{15}{1}$. Therefore, $\sqrt{225}$ is a rational number.