Square Root

Square root of a number is essentially the value that, when multiplied by itself, yields the original number. This concept is denoted by the radical symbol (√) and is expressed as √n or n^1/2, where ‘n’ is a positive number.

In this article, we will learn about, Square Root Definition, Symbol, Properties, Examples, and others in detail.

What is a Square Root?

A square root of a number is what you multiply by itself to get the original number. Let’s take the number 16. If you multiply 4 by itself, you get 16 (4 × 4 = 16). In this case, we say 4 is the square root of 16. The exponent for squares is 2, and for square roots, it’s 1/2. So, the square root of a number n is written as √n or n^1/2, where n is a positive number.

Square Roots Definition

A square root of a number is a value that, when multiplied by itself, equals the original number. If we have a positive integer m, the square root is denoted as √(n × n) or √(n²), which simplifies to n.

Symbol of Square Root

Symbol for Square Root is commonly written as √ and it’s known as a radical symbol. When we use this symbol to express a number ‘x’ as a square root, it looks like √x. Here, ‘x’ is the number. The actual number under the radical symbol, in this case, ‘x’, is called the radicand. The image added below show square root symbol,

For instance, if we have the square root of 4, we can also write it as the radical of 4, and both expressions indicate the same value.

Formula of Square Root

Formula for Square Root of a number x is written as √x or x^1/2.

This formula represents a mathematical operation that gives us a value, which, when multiplied by itself, equals the original number x. In simpler terms, if you have a number x, the square root is the value that, when squared, gives you back x.

For example, if y= √25, it means (y) is the number that, when multiplied by itself ( y × y), equals 25. In this case, (y = 5) because 5 × 5 = 25.

The alternative notation x^1/2 is another way to express the square root, indicating that we’re raising (x) to the power of 1/2, which is equivalent to taking the square root.

Properties of Square Root

Various Properties of Square Root are,

• If a number is a perfect square, it has a perfect square root.
• If a number has an even number of zeros at the end, it can have a square root.
• You can multiply two square roots together. For example, if you multiply √3 by √7, the result is √21.
• Multiplying two identical square roots gives a non-square root number. For instance, √4 multiplied by √4 equals 4.
• The square root of negative numbers is not defined because perfect squares cannot be negative.
• If a number ends with 2, 3, 7, or 8 in the unit digit, it does not have a perfect square root.
• If a number ends with 1, 4, 5, 6, or 9 in the unit digit, it may have a perfect square root.

How to Find Square Root?

To find the square root of a number, we consider which number, when squared, equals the given number. Finding the square root is straightforward for perfect squares, which are positive numbers expressed as the product of an integer by itself (or the value of an integer raised to the power of 2).

There are four methods to find the square root of numbers:

• Square Root by Prime Factorization Method
• Square Root by Estimation Method
• Square Root by Long Division Method
• Square Roots by Repeated Subtraction Method

Square Root by Prime Factorization Method

To determine the square root of a number using the prime factorization method, follow these steps:

Step 1: Break down the given number into its prime factors.

Step 2: Group the factors into pairs, ensuring that both factors in each pair are the same.

Step 3: Select one factor from each pair.

Step 4: Multiply the chosen factors together.

Step 5: The result of this multiplication represents the square root of the given number.

Example: Find the square root of 324 by Prime Factorization Method.

324 = 2² × 3⁴

Since we are finding the square root, pair the prime factors in twos: 2 × 3²

Multiply the prime factors: 2 × 3² = 2 × 9 = 18

The square root of 324 is √324 = 18

So, √324 = 18 using the prime factorization method.

Square Root by Estimation Method

Estimation and approximation involve making a reasonable guess of the actual value to simplify calculations and make them more realistic. This approach proves helpful in estimating the square root of a given number. Let’s apply this method to find √21. Identifying the nearest perfect square numbers to 21, we find 16 and 25. Knowing that √16 = 4 and √25 = 5, we can determine that √21 falls between 4 and 5.

Now, to refine our estimate, we evaluate the midpoint between 4 and 5, which is 4.5. Squaring 4.5 gives 20.25, and squaring 5 gives 25. Since 21 lies between these values, we can conclude that √21 is closer to 4.5. Therefore, our estimated square root for 21 is approximately 4.5.

Example: Estimate √65

Identify the perfect squares closest to 65. The perfect square smaller than 65 is 64 (8²) and the perfect square larger than 65 is 81 or 9².

Since 65 is closer to 64, start with the square root of 64, which is 8.

Determine whether you need to increase or decrease your estimate based on how far 65 is from the perfect squares.

• 8² = 64, which is less than 65.
• 9² = 81, which is more than 65.

Since 65 is closer to 81, increase the estimate.

Take the average of 8 and 9: (8+9)/2 = 8.5

Test (8.5²): 8.5 × 8.5 = 72.25

Since 72.25 is less than 65, further increase the estimate.

• Another average: (8.5+9)/2 = 8.75
• Test (8.75²): 8.75 × 8.75 = 76.5625

Since 76.5625 is greater than 65, stick with the previous estimate.

The estimated square root of 65 is approximately 8.75

This estimation method provides a reasonable approximation of the square root, especially when the number is not a perfect square.

Square Root by Long Division Method

Long division method for finding the square root of a number involves a series of steps that are performed similar to long division.

Read More, Square Root by Long Division Method

Here’s a step-by-step explanation of the method:

Example: Find the square root of 3249

Start from the right and group the digits into pairs, placing a bar over the pairs. For 3249, you have 32 and 49.

√ | (32 29)

Step 1: Determine the largest single-digit number whose square is less than or equal to the first pair on the left. In this case, it’s 5 because (5² = 25), which is less than 32.

Step 2: Subtract 25 (the square of 5) from 32, and bring down the next pair (49).

Step 3: Double the root (5) to get 10, and put an empty digit placeholder next to it.

Step 4: Guess a digit to fill in the empty placeholder (in this case, 4). Place it next to the previous result to make a new number (70).

Step 5: Multiply the entire result (10) by the guessed digit (4), and subtract from the dividend (70). Bring down the next pair (49).

Repeat steps 4 and 5 until you find the desired level of precision or until there are no more decimal places in the original number.

The process continues with more decimal places as needed. The square root of 3249 is approximately 56.

Square Roots by Repeated Subtraction Method

In this method, we repeatedly subtract consecutive odd numbers from the given number until we reach 0. The count of these subtractions indicates the square root of the original number. It’s important to note that this method specifically applies to perfect square numbers.

Example: Finding the square root of 25 using Repeated Subtraction Method

⇒25 – 1 = 24

⇒24 – 3 = 21

⇒21 – 5 = 16

⇒16 – 7 = 9

⇒9 – 9 = 0

Now we have subtracted the number five times. Therefore, √25 = 5

Square Root of Perfect Squares

The square root of a perfect square is a number that, when multiplied by itself, equals the original perfect square. For example, if a is a perfect square, then its square root is denoted as √ and √a × √a = a.

Example: Find the square root of 81.

Number 81 is a perfect square because it can be expressed as the product of an integer multiplied by itself: 9 × 9 = 81

The square root of 81 is the number that, when multiplied by itself, equals 81. In this case, the square root of 81 is 9, because 9 × 9 = 81.

Therefore, √81 = 9

In general, if (a² = b), then √b = a. Perfect squares have whole number square roots.

Table of Square Root

The table of first 50 natural numbers is added in the table below,

Square Root from 1-50
√1	1	√11	3.32	√21	4.58	√31	5.57	√41	6.4
√2	1.41	√12	3.46	√22	4.69	√32	5.66	√42	6.48
√3	1.43	√13	3.61	√23	4.8	√33	5.75	√43	6.56
√4	2	√14	3.74	√24	4.9	v34	5.83	√44	6.63
√5	2.24	√15	3.87	√25	5	√35	5.92	√45	6.71
√6	2.45	√16	4	√26	5.1	√36	6	√46	6.78
√7	2.65	√17	4.12	√27	5.2	√37	6.02	√47	6.86
√8	2.83	√18	4.24	√28	5.29	√38	6.16	√48	6.93
√9	3	√19	4.36	√29	5.39	√39	6.24	√49	7
√10	3.16	√20	4.47	√30	5.48	√40	6.32	√50	7.07

Expressions of Square Root

Square root can be expressed in different form such as, radical form, exponential form, decimal form, rationalizing form and the exact form.

Radical Form of Square Root

In radical form, the square root is expressed using the radical symbol (√). For example, the square root of 25 is written as √25, representing a value whose square is 25. This form is useful for indicating the principal square root of a non-negative number.

Exponential Form of Square Root

The exponential form represents the square root as an exponent. For instance, the square root of a number a can be expressed as^1/2 or a^0.5. This form connects the concept of square roots with exponentiation.

Decimal Form of Square Root

In decimal form, the square root is represented as a decimal. Calculators are often used to obtain accurate decimal approximations of square roots. For example, the square root of 16 in decimal form is 4. This form is practical for numerical computations.

Rationalizing Denominator

Rationalizing the denominator involves adjusting a fraction to eliminate square roots in the denominator. This is done by multiplying both the numerator and denominator by the conjugate of the denominator. The goal is to express the square root in a simplified and rationalized form.

Exact Form of Square Root

The exact form represents square roots without converting them to decimals. For instance, the square root of 2 is often left as √2 instead of providing a decimal approximation. This form is preferred when precision or simplicity in representation is necessary.

Square Root of Negative Number

When dealing with square roots, it’s important to note that the square root of a negative number doesn’t result in a real number. This is because squaring a number always yields a positive number or zero. However, complex numbers come into play to handle the square root of negative numbers. The principal square root of -x is expressed as √(-x) = i√x, where ‘i’ denotes the square root of -1.

Let’s consider an example with the perfect square number 9. Now, if we look at the square root of -9, there isn’t a real number solution. Expressing it using the complex number approach, √(-9) becomes √9 × √(-1), resulting in 3i {given that √(-1) = i}. In this way, 3i serves as a square root of -9.

Square Root of Complex Numbers

Complex numbers, which have both a real and an imaginary part, finding the square root involves considering each part separately.

The square root of a complex number (a + bi) (where (a) and (b) are real numbers, and (i) is the imaginary unit, (i = √-1) can be expressed using the following formula:

Here, sgn(b) represents the sign function, which gives -1 if (b) is negative, 0 if (b) is zero, and 1 if (b) is positive.

For example, Complex Number (4 + 3i)

Step 1: Find the magnitude of the complex number,

Step 2: Apply the formula for the square root of a complex number:

So, the square roots of the complex number (4 + 3i) are

Square Root of Decimal Number

A decimal is the value denoted by a (·), such as 0.9, 0.234, 0.14, etc.

For example to find the square root of 0.25. Let’s denote it as N and set N² equal to 0.25.

N² = 0.25

Now, taking the square root on both sides:

N = ±√0.25

0.5 × 0.5 = (0.5)² = 0.25

Therefore,

N = ±√(0.5)²

So, the square root of 0.25 is ±0.5.

Simplifying Square Root

Simplifying square roots involves expressing them in a simpler form by identifying perfect square factors.

For example the square root of 72. It can be written as:

√72

To simplify this, we break down 72 into its prime factors:

72 = 2 × 2 × 2 × 3 × 3

Now, we group the prime factors into pairs of identical numbers because the square root of a perfect square is a whole number. In this case, we have two pairs of 2s and one pair of 3s:

√72 = √(2 × 2 × 2) × (3 × 3)

Grouping these pairs:

√72 = 2 × 3 ×√2

Finally, multiplying the numbers outside the square root:

√72 = 6√2

So, the simplified form of the square root of 72 is 6√2. This process helps express square roots in a more concise and manageable way.

Square of a Number

When we square a number, we’re essentially multiplying the number by itself. If we have a number, ‘N,’ the square of that number is denoted as (N²), and it is found by multiplying ‘N’ by ‘N’.

For example if we take the number 4 and square it, the calculation is 4 × 4, which equals 16. So, 4² = 16. Similarly, for a number like 7, squaring it means (7 × 7), resulting in 49 (7² = 49).

In general terms, if we have any number ‘N’, squaring it is represented as (N × N), or (N²). The result is always a non-negative value, as the square of any real number is either zero or a positive value.

Differences between Squares and Square Roots

The difference between Square and Square Root is covered in the table added below,

Squares vs Square Roots
Squares	Square Roots
Product of a number multiplied by itself is called square of the number	Square root of number is number which when multiplied by itself gives the original number.
92 = 81	√(81) = 9
Exponent Symbol (^)	Square Root Symbol (√)
It is always positive	It can be either positive or negative
Inverse of Square is Square Root	Inverse of Square Root is Square

Examples of Square Root

Some examples of square roots are,

Example 1: Find the square root of 72.

Solution:

Step 1: Divide 72 by the smallest prime number, which is 2. (72 ÷ 2 = 36)

Step 2: Continue dividing until you can’t divide by 2 anymore. (36 ÷ 2 = 18, 18 ÷ 2 = 9)

Step 3: Move on to the next prime number, which is 3. (9 ÷ 3 = 3)

Step 4: The remaining number, 3, is a prime number itself.

So, the prime factorization of 72 is (2³ × 3²).

Find the square root using the prime factorization method:

Group the factors in pairs: (2 × 2 × 2) and (3 × 3).

Take one factor from each pair: 2 × 3 = 6

Therefore, the square root of 72 is 6.

Example 2: Find the square root of 16.

Solution:

16 – 1 = 15

15 – 3 = 12

12 – 5 = 7

7 – 7 = 0

Thus, the square root of 16 is 4.

Square Root – Practice Questions

1. Find the square root of 81.

2. Determine the square root of 144.

3. If (x² = 64), what is the value of (x)?

4. What is the square root of 25?

5. Solve for (y) if √y = 9

Square Root-FAQs

1. What is Square Root?

Square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the symbol.

2. How to Find Square Root of Perfect Squares?

To find the square root of perfect squares, you can use methods such as prime factorization, repeated subtraction, or estimation. Perfect squares are numbers that result from multiplying an integer by itself.

3. What are Squares and Square Roots?

Squares are the result of multiplying a number by itself. The square root is the inverse operation, representing the value that, when multiplied by itself, gives the original number. For example, 9 is the square of 3, and the square root of 9 is 3.

4. How to Find the Square Root of a Decimal Number?

To find the square root of a decimal number, use the same principles as with whole numbers. You can apply methods like prime factorization or use a calculator for more precise results. The decimal part is handled similarly to whole numbers.

5. Which Method is Used to Find Square Root of Non-Perfect Square Numbers?

Long division method is commonly used to find the square root of non-perfect square numbers. This method is versatile and works for any number, whether it’s a perfect square or not. Other methods like estimation and approximation can also be employed for non-perfect square roots.

6. What is Square Root Symbol?

Symbol of square root is called the radical sign ‘√‘.

7. What is Square Root Calculator?

Square Root Calculator is an online tool that is used to find the square root of any number. It is provided at our website.