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To estimate square roots, it's essential to understand the concept of perfect squares. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be written as 4 \( \times \) 4. Similarly, 25 is a perfect square because 5 \( \times \) 5 equals 25.
Identifying perfect squares helps us locate our target number between these squares, facilitating an easier estimation of its square root.
Here are some commonly known perfect squares to remember:

• 4, which is \( 2^2 \)
• 9, which is \( 3^2 \)
• 16, which is \( 4^2 \)
• 25, which is \( 5^2 \)
• 36, which is \( 6^2 \)

Recognizing these makes it easier to estimate square roots for other numbers.

A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 25 is 5 because 5 \( \times \) 5 equals 25. In mathematical terms, this is denoted as \( \sqrt{}25 = 5 \).
When working with square roots of non-perfect squares, we often need to estimate. To do this, we locate the number between two perfect squares and use their roots for our estimation. For example:

• The number 20 lies between the perfect squares 16 (\( 4^2 \)) and 25 (\( 5^2 \))
• Hence, \( 4 \lt \sqrt{}20 \lt 5 \)

This method helps us get close to the actual square root without needing a calculator.

Estimation is an invaluable skill, especially when dealing with square roots of numbers that are not perfect squares. To estimate the square root of a given number, follow these steps:

Identify the closest perfect squares:
Find two perfect squares that the given number falls between. For example, for 200, the nearest perfect squares are 196 (\( 14^2 \)) and 225 (\( 15^2 \)).

Calculate their square roots:
The square roots are 14 and 15, respectively, for 196 and 225. Thus, \( \sqrt{}200 \) lies between 14 and 15.

Refine your estimate:
Recognize whether the given number is closer to one perfect square than the other. Here, 200 is closer to 196 than to 225, so we can say \( \sqrt{}200 \) is closer to 14.1.
Practicing these steps will improve your estimation skills and make it easier to work with square roots without precise calculation.