Square Root: Formulas, Solved Examples, How to Find Square Root

Square Root: The square root of a number is a value that, when multiplied by itself, gives back the original number. For example, the square root of 16 is 4 because 4 x 4 = 16. Learning about square roots is very important in math because they are used in many areas like geometry, algebra, and real-life situations.

Knowing how to find square roots helps with solving problems related to area and volume and even working with equations. For junior students, understanding square roots lays a good foundation for learning more complex math later on.

In the following notes, we provide simple explanations, formulas, and examples to help you understand square roots better and make learning easier. Check out how to calculate square roots and learn through easy examples to improve your math skills.

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What is Square Root?

Imagine you have a number, like 9. The square root of 9 is a number that, when multiplied by itself, gives you 9. In this case, that number is 3, because 3 x 3 = 9.

A square root is the reverse process of squaring a number. Squaring a number means multiplying it by itself. A square root is finding the number that, when squared, gives you the original number. For example, the partner of 9 is 3, because 3 times 3 equals 9. The square root of a number x is a number y such that y² = x.

The square root symbol is denoted by '√'.

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Square Root Formula With Example

The square root of a number is the value that, when multiplied by itself, produces the original number. In other words, it's the reverse of squaring a number. Squaring involves multiplying a number by itself, while the square root determines the number that, when multiplied by itself, equals the given value.

The square root is shown with this symbol: '√'. For example, √9 means the square root of 9.

Square Root Formula:

If 'x' is the square root of 'y', we can write it like this:

x = √y

Or, equivalently:

x² = y

It means that when you square 'x', you get 'y'.

Square Root Example:

Let's take the number 16 and find its square root.

We write it as: √16

Now, we need to think of a number that, when multiplied by itself, gives 16.

In this case, 4 * 4 = 16. So, the square root of 16 is 4.

Therefore, √16 = 4.

This means 4 is the number that, when multiplied by itself, gives 16.

Square Root Symbol

The square root symbol (√) is a special symbol used in math to find the square root of a number. It looks like a checkmark with a line over it. When you see this symbol, it’s a sign that you need to find a number that, when multiplied by itself, gives you the original number. This process is the opposite of squaring a number.

Squaring means multiplying a number by itself, like 3 × 3 = 9. The square root is the reverse of this. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9.

So, whenever you see the square root symbol, think of it as asking you to find the number that, when multiplied by itself, gives you the original number. The square root symbol helps make this easy to represent in math problems.

Square Root 1 to 30

The table below shows the square roots of numbers from 1 to 30 along with their approximate values. This table will help you understand how to calculate square roots and get a better idea of what these values look like.

Knowing how to find square roots is an important skill in math, and this table can be a useful guide to help you practice and improve your understanding of this concept. By using the table, you can easily see how the square root of different numbers increases as the numbers get larger.

How to Find Square Root?

Finding the square root of a number can be done in different ways depending on the situation. Below are several methods that can be used to find the square root, each with its steps and examples to help you understand the process better.

1. Prime Factorization Method

This method is particularly useful for perfect squares (numbers that have whole number square roots).

Steps:

• Prime Factorization: Start by breaking the number down into its prime factors (numbers that are only divisible by 1 and themselves).
• Pair the Factors: Group the prime factors into pairs of the same number.
• Multiply One Factor from Each Pair: Take one factor from each pair and multiply them together. The result is the square root.

Example: Let’s find the square root of 144.

• Prime factorization of 144 is: 2 × 2 × 2 × 2 × 3 × 3.
• Pairing the factors: (2 × 2) × (2 × 2) × (3 × 3).
• Multiply one factor from each pair: 2 × 2 × 3 = 12.

Thus, the square root of 144 is 12.

2. Repeated Subtraction Method

This method works well for smaller perfect squares and involves subtracting odd numbers one by one.

Steps:

• Start Subtracting Odd Numbers: Begin by subtracting 1, then 3, 5, 7, and so on, from the number.
• Count the Subtractions: The number of times you subtract is the square root of the number.

Example: Let’s find the square root of 25.

• 25 - 1 = 24
• 24 - 3 = 21
• 21 - 5 = 16
• 16 - 7 = 9
• 9 - 9 = 0

We subtracted 5 times, so the square root of 25 is 5.

3. Long Division Method

This method works for both perfect and imperfect squares and is more complex, but it gives an accurate result.

Steps:

• Pair the Digits: Start by grouping the digits of the number into pairs, starting from the decimal point, both to the left and right.
• Find the Largest Square: Look for the largest perfect square less than or equal to the first pair of digits.
• Subtract and Bring Down the Next Pair: Subtract the square from the first pair and bring down the next pair of digits.
• Double the Divisor: Double the number you got in the quotient, and use it as the new divisor.
• Guess the Next Digit: Find a digit to place in the quotient so that the product of the new divisor and the new digit is less than or equal to the current dividend.
• Repeat: Continue this process until you have used all the digits.

Example: Let’s find the square root of 436.

1. Start by grouping the digits: (43) and (6).
2. The square root of 43 is 6, because 6 × 6 = 36.
3. Subtract 36 from 43, leaving 7, and bring down the next pair (06), making it 706.
4. Double 6 (the quotient so far) to get 12.
5. Guess the next digit to get 125 as the divisor. Multiply 125 by 5 (the next digit) to get 625, which is less than 706.
6. Subtract 625 from 706, leaving 81. There are no more digits to bring down.

The square root of 436 is approximately 20.7.

4. Estimation Method

For non-perfect squares, this method helps you get an approximate value for the square root.

Steps:

• Find the Nearest Perfect Square: Identify the perfect square closest to the number.
• Estimate the Square Root: Take the square root of the perfect square as your initial guess.
• Refine the Estimate: Use trial and error or a calculator to improve your guess.

Example: Let’s find the square root of 70.

• The nearest perfect square is 64, which has a square root of 8.
• Since 70 is a little more than 64, the square root of 70 will be a little more than 8.
• Using a calculator, we find that the square root of 70 is approximately 8.37.

Remember, for non-perfect squares, the square root will often be an irrational number, meaning it cannot be exactly expressed as a fraction. In such cases, we typically use a decimal approximation.

Natural Numbers

Difference Between Squares and Square Roots

Squares and square roots are closely related mathematical operations. Squaring a number means multiplying it by itself, while the square root of a number is the value that, when multiplied by itself, gives the original number. Below is a comparison between squares and square roots: