Result of Squaring an Integer	Perfect square is result of multiplying an integer by itself.
Negative Numbers Can Form Perfect Squares	Negative integers can form perfect square, e.g., (−4)2 = 16
Unique Square for Each Integer	Each integer has not a unique square. Two integers have one square, i.e. ‘a’ and ‘-a’ have same square.
Zero is a Perfect Square	Zero is considered a perfect square because 02 = 0
Sum of Consecutive Odd Numbers	A perfect square is sum of consecutive odd numbers.
Geometric Representation	Perfect square represents area of any figure.

Perfect Square Chart

Chart for Perfect Square is added below as,

Perfect Square – Tips and Tricks

Some Tricks and Tips for Perfect Squares are given below.

Square of a Number Ending in 5: To find square of a number ending in 5, multiply the digit before 5 with next digit and append 25. For example, 75²= 7×8(25) = 5625

Square of Numbers Close to 100: For numbers close to 100, express the square as (100 – x)²= 100² – 200x + x². This simplifies calculations, especially for mentally calculating squares.

Odd Number Squares: Square of any odd number is an odd number. If n is an odd number, then n² is odd.

Even Number Squares: Square of any even number is an even number. If m is an even number, then m² is even.

Difference of Squares: Use difference of squares formula, a²− b²= (a+b)(a−b). This can help in factoring or simplifying expressions.

Square of a Sum: (a+b)² = a² + 2ab + b²

Square of a Difference: (a−b)² = a² − 2ab + b²

Observations on Perfect Squares

Perfect numbers end with any one of these digits 0, 1, 4, 5, 6, or 9. Also some observations on perfect squares are,

• Numbers ending with 3 and 7 have 9 as units place digit in their square number.
• Numbers ending with 5 have 5as units place digit in their square number.
• Numbers ending with 4 and 6 will have 6 as units place digit in their square number.
• Numbers ending with 2 and 8 will have 4 as units place digit in their square number.
• Numbers ending with 1 and 9 will have 1 as units place digit in their square number.

How many Perfect Squares are between 1 and 100?

There are 8 perfect squares between 1 and 100 (excluding 1 and 100). They are,

4, 9, 16, 25, 36, 49, 64 and 81

How many Perfect Squares are between 1 and 1000?

There are 30 perfect squares between 1 and 1000. They are,

4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900 and 961

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Perfect Square Examples

Example 1: Identify the first two perfect squares.

Solution:

First two perfect squares are obtained by squaring the first two whole numbers:

• 1²=1 (Square of 1 is 1)
• 2²= 4² (Square of 2 is 4)

Therefore, first two perfect squares are 1 and 4.

Example 2: If a number is a perfect square and its square root is 9, what is the number?

Solution:

If a number is a perfect square and its square root is 9, we can find the number by squaring the square root:

9² = 81

So, required number is 81, as it is a perfect square, and its square root is 9.

Example 3: If a number is a perfect square and its square root is a prime number, find the number.

Take the prime number 5. The square of 5 is 25 (5²=25). Here, 25 is a perfect square, and 5 is a prime number.

So, the number we’re looking for is 25, where the square root (5) is a prime number

Practice Questions on Perfect Square

Some questions on perfect square are,

Q1: Find the square of 5.

Q2: Is 36 a perfect square?

Q3:. Determine the square root of 49.

Q4: Write next two perfect squares after 16.

Q5: Identify the perfect square closest to 150.

FAQs on Perfect Square

How many Perfect Squares are between 1 and 100?

There are 10 perfect squares between 1 and 100. These are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

How many Perfect Squares are between 1 and 1000?

There are 31 perfect squares between 1 and 1000. These include numbers like 1, 4, 9, 16, 25, and so on, up to 961.

Is 216 a perfect square?

Yes, 216 is a perfect square. The square root of 216 is 14, because 14 multiplied by itself (14 × 14) equals 216.

What defines a perfect square?

A perfect square is a number that can be made by multiplying a whole number by itself. For instance, 9 is a perfect square because it’s 3 times 3.

How does one determine if a number qualifies as a perfect square?

To check if a number is a perfect square, you see if it can be expressed as the product of a whole number multiplied by itself. If yes, it’s a perfect square.

In mathematical terms, what characterizes a perfect square trinomial?

A perfect square trinomial in math is an expression that can be factored into two identical binomials. It has the form (a+b)².

Which numerical values are considered perfect squares?

Numbers like 1, 4, 9, 16, and so on, are perfect squares. They result from multiplying a whole number by itself.

What is the process for factoring perfect squares?

To factor perfect squares, you write them as the square of a binomial. For example, 25=(5)²

What approach is used to identify perfect squares?

Identifying perfect squares involves finding if a number can be written as the product of a whole number multiplied by itself.

Does the number 7 qualify as a perfect square?

No, 7 is not a perfect square. You can’t get it by multiplying a whole number by itself.