Square and Square Roots - Formulas, Properties, and Methods

Table of Contents

When you take a number and multiply it by itself, you get what’s called the “square of that number. To show this, we use a little 2 as an exponent, like this: a² means “the square of a.

Now, the square root of a number is a value that, when multiplied by itself, gives you back the original number. We represent it using a special symbol that looks like a checkmark, like this: √a means the square root of a.

For instance, if you take the number 5 and square it (5²), you get 25. And if you take the square root of 25 (√25), you get 5.

Properties of Square Numbers

The square of a negative number becomes positive: (-a)² = a²
The square of a positive number remains positive: (a)² = a²
A positive number’s square root has two real solutions, one positive and one negative: √25 = 5 or -5
The square root of 0 is 0.
The square root of a negative number results in a complex number.

These fundamental properties are crucial for understanding square numbers and their roots in mathematics, as well as in various real-life applications.

Perfect Squares

Perfect squares are numbers that are the result of an integer multiplied by itself. In other words, a perfect square is the square of an integer. For example, 1, 4, 9, 16, and 25 are perfect squares because they are the results of 1², 2², 3², 4², and 5², respectively. In general, the square of an integer (n) is denoted as (n²). Perfect squares have various applications in mathematics, including in the study of number theory, geometry, and algebra.

Square Roots of Perfect Squares

Perfect Squares	Square Root (√)
0	0
1	1
4	2
9	3
16	4
25	5

Imperfect Squares

Imperfect squares are numbers that cannot be obtained by multiplying a whole number by itself. In simpler terms, they are numbers that do not have a perfect square root, which means they are not the result of squaring an integer. For example, 2, 3, 5, 6, and 7 are imperfect squares because they cannot be expressed as the product of an integer multiplied by itself.

Square Root of Imperfect Squares

√5 ≈ 2.2361
√10 ≈ 3.1623
√13 ≈ 3.6056
√18 ≈ 4.2426
√20 ≈ 4.4721
√26 ≈ 5.0990
√29 ≈ 5.3852
√45 ≈ 6.7082
√50 ≈ 7.0711

Methods of Finding Square Root

Finding the square root of a number involves determining a value that, when multiplied by itself, gives the original number. There are several methods for finding square roots,

Prime Factorization Method
Long Division Method

Prime Factorization Method

This method involves breaking down the number into its prime factors and then pairing them to find the square root.

How it works:

Factorize the number into its prime factors.
Pair the prime factors. If there’s an odd number of a particular factor, leave one out of a pair.
For each pair, take one factor out of the square root sign.
Multiply these factors. This is the square root of the perfect square. For imperfect squares, an unpaired prime factor will remain under the square root.

Example: Find the square root of 36.

Prime factorization of 36: 2 × 2 × 3 × 3
Pair the factors: (2 × 2) and (3 × 3)
Take one factor from each pair: 2 and 3
Multiply these factors: 2 × 3 = 6
So, √36 = 6

Long Division Method

This is a systematic approach to find the square root of larger numbers.

How it works:

Divide the number into pairs of digits, starting from the decimal point (or the unit digit if there’s no decimal).
Find a number whose square is less than or equal to the first pair. This number is the first divisor and also the first digit of the square root.
Subtract the square of this number from the first pair and bring down the next pair.
Double the current quotient. This doubled number is part of the next divisor.
Find a digit which, when added to the doubled number and multiplied by the same digit, results in a product less than or equal to the current number.
Repeat the process until you reach a satisfactory level of precision.

Example: Find the square root of 1225.

Pair the digits: 12 | 25
The largest square less than 12 is 9 (3^2), so 3 is the first digit of the root.
Subtract 9 from 12, and bring down 25 to get 325.
Double the quotient (3) to get 6. The next digit of the root is found by figuring out what digit (x) in the 60x (which stands for 60 + x) that makes 60x × x ≤ 325.
x is 5 because 65 × 5 = 325.
So, √1225 = 35.

Squares of Negative Numbers

The squares of negative numbers behave just like the squares of positive numbers, but there’s a crucial distinction: the square of any negative number is consistently positive. This happens because when you multiply a negative number by itself, the two negative signs negate each other, giving you a positive result. In mathematical terms, (-x)² always equals x², where x is a positive number.

Mathematical Explanation:

The square of a number n is defined as n × n.
For a negative number, say -a (where a is positive), the square is (-a) × (-a).
A negative times a negative gives a positive, so (-a) × (-a) = a × a, which is positive.

Examples:

Square of -2: Calculation: (-2) × (-2), Result: 4
Square of -5: Calculation: (-5) × (-5), Result: 25
Square of -3.5: Calculation: (-3.5) × (-3.5), Result: 12.25

Square Roots 1 to 50

Square Roots 1 to 50
1² = 1	11² = 121	21² = 441	31² = 961	41² = 1681
2² = 4	12² = 144	22² = 484	32² = 1024	42² = 1764
3² = 9	13² = 169	23² = 529	33² = 1089	43² = 1849
4² = 16	14² = 196	24² = 576	34² = 1156	44² = 1936
5² = 25	15² = 225	25² = 625	35² = 1225	45² = 2025
6² = 36	16² = 256	26² = 676	36² = 1296	46² = 2116
7² = 49	17² = 289	27² = 729	37² = 1369	47² = 2209
8² = 64	18² = 324	28² = 784	38² = 1444	48² = 2304
9² = 81	19² = 361	29² = 841	39² = 1521	49² = 2401
10² = 100	20² = 400	30² = 900	40² = 1600	50² = 2500

FAQs on Square and Square Roots

Is square equal to square root?

No, a square and a square root are not equal. Squaring a number means multiplying it by itself, while taking the square root of a number means finding a value that, when squared, gives the original number.

How do you find square √?

To find the square root (√) of a number, you can use methods like prime factorization, long division, or a calculator. The square root is the number that, when multiplied by itself, equals the original number.

What is square formula?

The square formula refers to squaring a number, which is done by multiplying the number by itself. Mathematically, its expressed as n^2, where n is the number being squared.

What is the square root of 400?

The square root of 400 is 20. This is because 20 multiplied by 20 equals 400. Hence, 20 is the number that, when squared, gives the value of 400.

Is 676 a perfect square?

Yes, 676 is a perfect square. The square root of 676 is 26, as 26 times 26 equals 676. A perfect square is a number that can be expressed as the square of an integer.