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A perfect square is a number that results from squaring an integer. In other words, it is the product of an integer multiplied by itself. For example, 1, 4, 9, 16, and 25 are perfect squares because they can be written as:

• 1 = 1 × 1
• 4 = 2 × 2
• 9 = 3 × 3
• 16 = 4 × 4
• 25 = 5 × 5

In our exercise, we are asked to evaluate the square root of 225. To determine if 225 is a perfect square, we try to find an integer that multiplies by itself to give 225. It turns out that 15 × 15 = 225, confirming that 225 is a perfect square. Therefore, \(\root{225}\) equals 15. If a number is a perfect square, its square root will always be an integer.

Not all numbers have integer square roots. When a number cannot be written as the product of two identical integers, it is not a perfect square. The square root of such a number is an irrational number. Irrational numbers cannot be expressed as a simple fraction or a repeating or terminating decimal. For example, \(\root{2}\) and \(\root{3}\) are irrational numbers because their decimal forms go on forever without repeating.
When it is not possible to simplify a square root to an integer, we usually approximate the value and round it to a specific decimal place, often the nearest hundredth.
It’s important to recognize whether a number under the square root sign is a perfect square to determine if its square root is rational (an integer or a simple fraction) or irrational.

To evaluate a square root, follow these steps:

• Identify the number under the square root sign.
• Check if the number is a perfect square by determining if it can be written as the square of an integer.
• If it is a perfect square, find the integer whose square equals the number.
• If it is not a perfect square, approximate the square root and round it to the desired decimal place.

Let’s review our exercise – \(\root{225}\). First, we identified that 225 is a perfect square since 15 × 15 equals 225. Therefore, the square root of 225 is exactly 15, which is an integer. No rounding is required. When dealing with non-perfect squares, remember to use a calculator for approximations and round as instructed.