Square root

Introduction to Square Root

The topic of square roots is an important part of mathematics and is used in many areas such as algebra, geometry, and measurements. The word 'square root' means the number when it's multiplied by itself. For example, the square root is 25 is 5 because 5 × 5 = 25.

Square roots are written with the symbol √   is called the 'radical sign'. A square root can be a whole number (e.g., √ 49 = 7) or a decimal (e.g., √ 2 ≈ 1.414). Each positive number consists of two square roots: one positive and one negative. For example, both 5 and -5 are square roots of 25. The zero has only one square root, which is 0.

In this article, we will learn more about square roots, their properties, how to calculate them, and their use in solving problems. We will also see step-by-step examples that will make it easier to understand and apply the concepts of square roots under real-life conditions and mathematics.

 

Table of Contents

• Square Roots Definition
• Different Methods to Find Square Root
• Repeated Subtraction Method
• Prime Factorization Method
• Estimation or Approximation Method
• Long Division Method
• Square Root of Perfect and Non-Perfect Squares
• Square Root Table
• How to Simplify Square Roots
• Square Root Formulas and Properties
• Practice Questions
• Conclusion
• FAQs on Square Roots

 

Square Roots Definition

In mathematics, it is necessary to understand the square roots and squares, especially when multiplication, algebra, and geometry. A square number is the result of multiplying a number by itself, while a square root is the value that gives the original number when squared. These two concepts are closely linked: square routing and routing reverse operations.

The square root of a number is a value that gives the original number when multiplied by itself. For example, if we take the number 9, the square root is 3 because 3 × 3 = 9.

In mathematics, the square root function takes a positive number as input and provides the square root as output.

We can write it this way:

f(x) = √ x

If 𝑥= 16, then f(x) =  √ 16 = 4

For example:  √ -n = i √ n. Here, 𝑖 am called an imaginary unit.

Square root symbol

The Square Root symbol is written as √  and it is called a radical symbol. The number inside the symbol is called the Radicand.

For example: √ 25 = 25

Different Methods to Find Square Root

There are several ways to calculate the square root of a number, each of which is compatible with different types of numbers and learning levels. While some repeated subtractions are simple and comfortable as a method, others as long division methods, provide more accuracy for large or non-perfect square numbers. Choosing the right method depends on whether the number is an ideal class and how much accuracy is required.

1. Repeated Subtraction Method

This class is one of the simplest ways to understand roots, especially for perfect square numbers. It continuously means to subtract odd numbers from the given number until you reach zero.

Steps:

1. Start subtracting odd numbers (1, 3, 5, 7, ...) from the given number.
2. Count the number of steps required to reach zero.
3. That count is the square root of the number.

Example:

Find √16 using repeated subtraction:

• 16 – 1 = 15
• 15 – 3 = 12
• 12 – 5 = 7
• 7 – 7 = 0
• Steps = 4 → √16 = 4

2. Prime Factorization Method

This method is commonly taught in middle school & is effective for finding square roots of perfect squares using their prime factors.

Steps:

1. Write the number as a product of its prime factors.
2. Pair the prime
3. Take one number from each pair & multiply to get the square root.

Example: Find √144

• Prime factorization: 144 = 2 × 2 × 2 × 2 × 3 × 3
• Grouping: (2 × 2), (2 × 2), (3 × 3)
• Take one from each: 2 × 2 × 3 = 12
• √144 = 12

3. Estimation or Approximation Method

This method helps to estimate the square root of numbers that are not perfect squares. This is useful when requiring quick and close answers.

Step:

1. Find the two perfect squares that the number is contained in.
2. Estimate the root by observing which perfect square it's closer to.
3. If necessary, process the estimate with average methods.

Example:

Find an approximate value for √50:

• Lies between √49 = 7 and √64 = 8
• Since 50 is closer to 49, √50 ≈ 7.1

4. Long Division Method

The Long Division method is an accurate way to find the square root of large numbers or decimal values. This is especially effective when the square root is not a whole number.

Steps:

1. In the couple starting from the decimal point, the group points (to the right of the decimal).
2. Find the largest number, if square. The first is the group.
3. Take down the next group, double the quota, and use it as a new divisor with the trial digit.
4. Repeat the process.

Comparison of All Square Root Methods

Square Root of Perfect and Non-Perfect Squares

Understanding the square root of Perfect and non-perfect squares is important for mastery in mathematics. A perfect square contains an accurate number in the form of the square root, while a non-perfect square has a decimal or an irrational number as a result. Differences between both parts and learning to work are important for students in the grade.

List of Perfect Squares from 1 to 100

A perfect square is are number that is the product of an integer with itself (e.g., 3 × 3 = 9)
Perfect Squares Between 1 and 100:

Square Root Table for Numbers 1 to 20

Here’s a helpful table to refer to when finding the square roots of numbers from 1 to 20, especially for both perfect & non-perfect squares.

Square Root Table:

Square Root of Non-Perfect Squares

Non-perfect square numbers are those whose square roots are not in full numbers. The square root of these numbers is always a decimal and usually irrational. This is the number 2, 3, 5, 7, etc.

How to identify non-perfect squares:

• They do not appear in the multiplication tables to any integer.
• Their square roots are not whole numbers.

Examples:

• √2 ≈ 1.414
• √5 ≈ 2.236
• √11 ≈ 3.317

 

How to Simplify Square Roots

Simplifying square roots means expressing them in their simplest radical form. It helps to reduce complex square root values ​​in a cleaner and more usable format. It is an important concept in algebra and high-level mathematics and is often used in equations, geometry, and trigonometry. There are many ways to simplify square roots, but the most common and reliable product and quotient rules use the main factor.

Using Prime Factorization

The Prime Factorization method is a widely used technique to simplify square roots. In this method, the number inside the square root has been added to break down into the most important prime factors and transfer numbers outside the radical.

Steps to simplify using Prime Factorization:

1. The factor number under the square root is in the most important components.
2. Identification and group added the same factors.
3. Transfer a factor from each pair outside the radical.
4. Multiply the numbers inside and outside the radical separately.

Example : Simplify √72 = 6√2

Let’s apply the above steps to simplify √72:

• Step 1: Prime factorization of 72 = 2 × 2 × 2 × 3 × 3
• Step 2: Group the pairs: (2 × 2) and (3 × 3)
• Step 3: Take one from each pair outside: 2 × 3 = 6
• Step 4: The Remaining factor inside the root = 2

Product and Quotient Rule of Square Roots

These are algebraic properties that help simplify more complex square root manifestations, especially when working with multiplication or division under the radical sign.

Product Rule:

• √a × √b = √(a × b)

• This rule allows us to multiply two square roots together easily.

• Example: √3 × √12 = √(3 × 12) = √36 = 6

Quotient Rule:

• √(a ÷ b) = √a ÷ √b(Only applicable when b ≠ 0)

• This rule allows us to divide the values under square roots individually.

• Example: √(49 ÷ 4) = √49 ÷ √4 = 7 ÷ 2 = 3.5

 

Square Root Formulas and Properties

Understanding the formulas and properties of square roots helps to simplify mathematical expressions and solve equations effectively. These formulas are used on a large scale in algebra, geometry, and real-world problems. Let's find out the most important square root formula and related properties that control the square roots, and how to behave under division & multiplication.

Square Root Formula

The formula to find the square root is:

y = √a

Since,

y * y = y² = a

Here, ‘a' is a number, and ‘y’ is its square root.

Example:
If a=36, then √36 = 6, because $6\times 6=36$.

Properties of Square Roots

Square roots follow a set of critical algebraic properties, especially when working with multiplication or division under the root. These are known as the product property and Quotient property of square roots.

√(a × b) = √a × √b

It is known as the product property of the square roots.

• This allows you to share or combine square roots during multiplication.
• This rule is especially useful when a number is an ideal square.

Example: √(4 × 9) = √4 × √9 = 2 × 3 = 6

√(a ÷ b) = √a ÷ √b

This is known as the Quotient property of square roots (valid when b ≠ 0)

• We can separate or add square roots during division.
• It helps simplify square root expressions involving fractions.

Example: √(36 ÷ 4) = √36 ÷ √4 = 6 ÷ 2 = 3

Practice Questions

Basic Level Square Root Problems

• Find √64
• Find √49
• Simplify √100
• Write 36 as the square of an integer
• Express √81 using exponential form

Application-Based Word Problems

• A square garden has an area of 144 square meters. What is the length of one side?
• The area of a square tile is 225 cm². Find the length of one side.
• A rope is cut into 64 equal square pieces. What is the side length of each square if the total length of the rope is √4096 cm?
• A square board has an area of 625 in². What is its perimeter?

Conclusion

Square roots are one of the basic concepts of mathematics. They help us understand the relationship between numbers, make calculations in areas such as geometry, algebra, engineering, and everyday life much simpler. Students can confidently solve the perfect and non-perfect square root problems by mastering methods such as frequent subtraction, prime factorisation, integrity, and long division. Understanding irrational numbers and knowing how to simplify square roots are important steps to achieve high mathematics skills. The more practice and application they have, the stronger their number sense becomes & allowing them to resist the challenges of mathematics with ease and accuracy.

FAQs on Square Roots

1. What does 5 √ 2 mean?

Ans: Thus, the square root of 50 is the value that is squared to get the original number. The simplified form of the square root of 50 is 5√2 or 7.07 (approximately). The square root of 50 can be represented in three forms. Radical form: √50 = 5√2.

2. What is the square root of 40?

Ans: The square root of 40 is symbolically expressed as √40. Thus, if we multiply the number 6.3245 two times, we get the original value 40. √40 = ± 6.3245. Square Root of 40 in Decimal Form: 6.3245.

3. How to solve √70?

Ans: The square root of 70 is expressed as √70 in the radical form and as (70)^1/2 or (70)^0.5 in the exponent form. The square root of 70, rounded up to 10 decimal places, is 8.3666002653. It is the positive solution of the equation x² = 70.

4. Is 29 a perfect square?

Now, 29 is not a perfect square number because we cannot find any integer that could be multiplied twice to get 29. Thus, we find an approximate value of the square root of 29, as it is an irrational number.

5. Does 2 have a square root?

Ans: Yes, 2 has a square root, which is approximately 1.414. The square root of 2 is an irrational number, meaning it cannot be expressed as a simple fraction, and its decimal representation continues infinitely without repeating.