# Perfect Squares

We may have encountered various numbers kinds in mathematics, including even, odd, prime, composite, etc. A perfect square is one particular kind of integer, though. These can be located and stated with the use of number **factorization**. On this page, you will discover the meaning of perfect square numbers, their notation, a list of these numbers between 1 and 100, and more.

## Perfect Squares Definition

“**Perfect squares are numbers that can be expressed as the square of an integer. In other words, a perfect square results from multiplying an integer by itself.**“

## Perfect Square Numbers

Perfect square numbers are numbers that can be expressed as the square of an integer. These numbers have an interesting property in that their square roots are also integers. The first few square numbers are as follows:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …

These sums are produced by multiplying each integer by itself. For instance, 1 is a perfect square since 1 x 1 equals 1 may be used to calculate it. Similar to 2, multiplying 2 by 2 results in 4 (2 x 2 = 4), and so forth. As a result, 4 is also a perfect square.

## Perfect Squares from 1 to 100

Perfect square numbers from 1 to 100 | ||||

1 | = | 1 × 1 | = | 1^{2} |

4 | = | 2 × 2 | = | 2^{2} |

9 | = | 3 × 3 | = | 3^{2} |

16 | = | 4 × 4 | = | 4^{2} |

25 | = | 5 × 5 | = | 5^{2} |

36 | = | 6 × 6 | = | 6^{2} |

49 | = | 7 × 7 | = | 7^{2} |

64 | = | 8 × 8 | = | 8^{2} |

81 | = | 9 × 9 | = | 9^{2} |

100 | = | 10 × 10 | = | 10^{2} |

## Perfect Squares List

1 = 1^{2} | 441 = 21^{2} | 1681 = 41^{2} |

4 = 2^{2} | 484 = 22^{2} | 1764 = 42^{2} |

9 = 3^{2} | 529 = 23^{2} | 1849 = 43^{2} |

16 = 4^{2} | 576 = 24^{2} | 1936 = 44^{2} |

25 = 5^{2} | 625 = 25^{2} | 2025 = 45^{2} |

36 = 6^{2} | 676 = 26^{2} | 2116 = 46^{2} |

49 = 7^{2} | 729 = 27^{2} | 2209 = 47^{2} |

64 = 8^{2} | 784 = 28^{2} | 2304 = 48^{2} |

81 = 9^{2} | 841 = 29^{2} | 2401 = 49^{2} |

100 = 10^{2} | 900 = 30^{2} | 2500 = 50^{2} |

121 = 11^{2} | 961 = 31^{2} | 2601 = 51^{2} |

144 = 12^{2} | 1024 = 32^{2} | 2704 = 52^{2} |

169 = 13^{2} | 1089 = 33^{2} | 2809 = 53^{2} |

196 = 14^{2} | 1156 = 34^{2} | 2916 = 54^{2} |

225 = 15^{2} | 1225 = 35^{2} | 3025 = 55^{2} |

256 = 16^{2} | 1296 = 36^{2} | 3136 = 56^{2} |

289 = 17^{2} | 1369 = 37^{2} | 3249 = 57^{2} |

324 = 18^{2} | 1444 = 38^{2} | 3364 = 58^{2} |

361 = 19^{2} | 1521 = 39^{2} | 3481 = 59^{2} |

400 = 20^{2} | 1600 = 40^{2} | 3600 = 60^{2} |

This allows us to deduce the equation for calculating the difference between any perfect square number and its forerunner. The equation yields this:

**n ^{2} − (n − 1)^{2} = 2n − 1**

However, the formula can be used to determine how many square numbers there are:

**n ^{2} = (n − 1)^{2} + (n − 1) + n**

## Perfect Squares Examples

In addition to the numerals, perfect square numbers can also be found in algebraic identities and polynomials. A factorization technique can be used to locate these.

Identities in algebra as perfect squares:

**a ^{2} + 2ab + b^{2} = (a + b)^{2}**

**a ^{2} – 2ab + b^{2} = (a – b)^{2}**

Polynomials as perfect squares:

Let us take the polynomial x^{2} + 10x + 25.

Now, factorize the polynomial.

**x ^{2} + 10x + 25 = x^{2} + 5x + 5x + 25**

= x(x + 5) + 5(x + 5)

= (x + 5)(x + 5)

**= (x + 5) ^{2}**

Let us take another example:

**x ^{2} – 12x + 36 = x^{2} – 6x – 6x + 36**

= x(x – 6) – 6(x – 6)

= (x – 6)(x – 6)

**= (x – 6) ^{2}**

From the above examples, we can say that x^{2} + 10x + 25 and x^{2} – 12x + 36 are called perfect square trinomials.

## Perfect Squares Chart

# Factorization of Perfect Squares

The factorization of a perfect square involves expressing it as a product of its prime factors, where each prime factor appears twice (since it is squared). Here’s how you can factorize a perfect square:

- Start with the perfect square number you want to factorize.
- Find the prime factorization of the perfect square number.
- Write down each prime factor twice, as the prime factorization of a perfect square will raise each prime factor to an even exponent.
- Multiply the prime factors together to get the factorization of the perfect square.

Let’s take an example to illustrate the process:

Example: Factorize the perfect square number 36.

- The number 36 is a perfect square.
- Find the prime factorization of 36: 36 = 2 * 2 * 3 * 3.
- Write each prime factor twice: 2 * 2 * 3 * 3.
- Multiply the prime factors: 2 * 2 * 3 * 3 = 2^2 * 3^2.

Therefore, the factorization of the perfect square number 36 is 2^2 * 3^2.

In general, the factorization of a perfect square will have each prime factor raised to an even exponent. It ensures that when the factors are multiplied, each factor contributes to the perfect square property.

By using two methods, we can do factorization of perfect squares which are as follows:

**By ****Using the square of a binomial pattern**

**Using the square of a binomial pattern**

The square of a binomial pattern is a special form of a polynomial that can be factored using the following formula:

(a + b)^2 = a^2 + 2ab + b^2To factor a perfect square using the square of a binomial pattern, follow these steps:

- Identify the binomial in the square.
- Find the square of each term in the binomial.
- Add the squares of the terms together.
- The factored expression is the square of the binomial plus the square of the terms.

Here are two examples of factoring perfect squares using the square of a binomial pattern:

**Example 1:**

**Factor the polynomial** **x^2 + 6x + 9**.

The binomial in the square is x + 3.

The square of x is x^2.

The square of 3 is 9.

Adding the squares together, we get x^2 + 9.

Therefore, the factored expression is **(x + 3)^2**.

**Example 2:**

**Factor the polynomial y^2 + 10y + 25**.

The binomial in the square is y + 5.

The square of y is y^2.

The square of 5 is 25.

Adding the squares together, we get y^2 + 25.

Therefore, the factored expression is **(y + 5)^2**.

## By **Using the difference of squares pattern**

The difference of squares pattern is a special form of a polynomial that can be factored using the following formula:

(a + b)(a – b) = a^2 – b^2To factor a perfect square using the difference of squares pattern, follow these steps:

- Identify the binomial in the difference.
- Find the square of each term in the binomial.
- Subtract the squares of the terms together.
- The factored expression is the square of the binomial minus the square of the terms.

Here are two examples of factoring perfect squares using the difference of squares pattern:

**Example 1:**

**Factor the polynomial x^2 – 9.**

The binomial in the difference is x – 3.

The square of x is x^2.

The square of 3 is 9.

Subtracting the squares together, we get x^2 – 9.

Therefore, the factored expression is **(x – 3)(x + 3)**.

**Example 2:**

**Factor the polynomial y^2 – 25.**

The binomial in the difference is y – 5.

The square of y is y^2.

The square of 5 is 25.

Subtracting the squares together, we get y^2 – 25.

Therefore, the factored expression is **(y – 5)(y + 5)**.

# Properties of Perfect Squares

The properties of perfect squares are given below:

## Multiplication Property

The product of two perfect squares is also a perfect square. For example if a and b are perfect squares then a * b is also a perfect square.

## Division Property

The quotient of two perfect squares is a perfect square. If a is a perfect square and b is a perfect square then a / b is also a perfect square [when the division is exact].

## Addition Property

The sum of two consecutive perfect squares forms a centered square number. For example 1 + 4 = 5, 4 + 9 = 13, 9 + 16 = 25 and so on.

## Difference Property

The sum of the corresponding odd numbers gives the difference between two consecutive perfect squares. For example 4 – 1 = 3 (1 + 2), 9 – 4 = 5 (3 + 2), 16 – 9 = 7 (5 + 2) and so on.

## Prime Factorization Property

The prime factorization of a perfect square consists of prime factors raised to even exponents. This property ensures that when the factors are multiplied, each factor contributes to the perfect square property.

## Geometry Property

Perfect squares have a connection to geometric shapes, particularly squares. A square’s area is a perfect square, and a square’s side length is equal to the square root of that area.

## Pythagorean Triples

**Pythagorean triples** which are collections of three positive numbers (a, b, and c) that satisfy the Pythagorean theorem (a2 + b2 = c2) depend on perfect squares. A and B make up a Pythagorean triple, while C is the square of another number.

# FAQ’s

### What is perfect square theory?

According to the perfect square theory a number is only a perfect square if and only if it can be stated as the sum of two identical elements. In other words, multiplying a number by itself produces a perfect square. For instance, the number 9 is a perfect square since it can be written as 3 x 3 (9). This theory aids in recognizing and using perfect square numbers in calculations and problem-solving in mathematics.

### Is 24 a perfect square?

No, 24 is not a perfect square. It cannot be expressed as the product of two identical factors. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24, none of which are the same. Therefore, 24 does not meet the criteria for a perfect square.

### Is 0 a perfect square?

Absolutely, 0 is a perfect square. Any number that can be written as the result of multiplying an integer by itself is referred to be a perfect square. It can be written as 0 times 0 (0 x 0 = 0), which satisfies the requirements for a perfect square in the case of 0. As a result, 0 is regarded as a perfect square.

### Why is 70 not a perfect square?

70 is not a perfect square because it cannot be expressed as the product of two identical factors or as the square of an integer. The factors of 70 are 1, 2, 5, 7, 10, 14, 35 and 70, none of which are the same. Therefore, there is no integer that can be multiplied by itself to yield 70, making it not a perfect square.

### Is negative 1 perfect square?

No, a perfect square is not -1. A non-negative number that may be written as the square of an integer is referred to as a perfect square. But whether a real number is positive or negative, its square will always have a non-negative value. The square of -1 is 1, hence it does not fit the definition of a perfect square since 1.

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