Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It is represented mathematically as √x, where x is the number.

For example, the square root of $9$ is $3$ because $3$ x $3 = 9.$

Square roots can be calculated using various methods, including the Babylonian method, long division, and the use of a calculator.

The square root of a number can be either a rational number (expressible as a fraction of two integers) or an irrational number (non-repeating, non-terminating decimal).

The square root of a negative number is not a real number and is represented as an imaginary number, denoted by the symbol i.

Square roots have applications in various fields, including mathematics, physics, engineering, and computer science.

Square Roots Definition

The square root of any number is equal to a number, which when squared gives the original number. Let us say m is a positive integer, such that $√(m.m) = √(m^2) = m$

In mathematics, a square root function is defined as a one-to-one function that takes a positive number as an input and returns the square root of the given input number.

$f(x) = √x$

Examples of Square Roots

Here are some examples of square roots:

• The square root of 1 is 1.
• The square root of 4 is 2.
• The square root of 9 is 3.
• The square root of 16 is 4.
• The square root of 25 is 5.

Square Root Symbol

The square root symbol is usually denoted as ‘√’. It is called a radical symbol. To represent a number ‘x’ as a square root using this symbol can be written as: ‘ √x ‘

where x is the number. The number under the radical symbol is called the radicand. For example, the square root of 6 is also represented as radical of 6. Both represent the same value.

Square Root Formula

The formula to find the square root is:

$y = √a$

Since, $y.y = y^2 = a$; where ‘a’ is the square of a number ‘y’.

Properties of Square root

The square root of a number is a number that, when multiplied by itself, produces the original number. For example, the square root of 9 is 3 because 3 x 3 = 9.

There are a number of properties of square roots that are useful to know. These properties can be used to simplify calculations and to solve problems.

1. The square root of a positive number is always positive.

This is because the square of a positive number is always positive. For example, the square root of 4 is 2 because 2 x 2 = 4.

2. The square root of a negative number is not a real number.

This is because the square of a negative number is always negative. For example, the square root of -4 is not a real number because there is no number that, when multiplied by itself, produces -4.

3. The square root of 0 is 0.

This is because 0 x 0 = 0.

4. The square root of 1 is 1.

This is because 1 x 1 = 1.

5. The square root of a fraction is the square root of the numerator divided by the square root of the denominator.

For example, the square root of 9/4 is 3/2 because 3/2 x 3/2 = 9/4.

6. The square root of a product is the product of the square roots.

For example, the square root of 9 x 4 is 3 x 2 = 6.

7. The square root of a quotient is the quotient of the square roots.

For example, the square root of 9/4 is 3/2 because 3/2 / 3/2 = 1.

8. The square root of a power is the power of the square root.

For example, the square root of $9^2$ is $3^2 = 9.$

9. The square root of a sum is not equal to the sum of the square roots.

For example, the square root of 9 + 4 is not equal to 3 + 2.

10. The square root of a difference is not equal to the difference of the square roots.

For example, the square root of 9 - 4 is not equal to 3 - 2.

These are just a few of the properties of square roots. There are many other properties that can be derived from these basic properties.

How do Find Square Root of Numbers?

the methods to find the square root of numbers are:

Square Root by Prime Factorisation

Square Root by Repeated Subtraction Method

Square Root by Long Division Method

Square root By Prime Factorization

The square root of a perfect square number is easy to calculate using the prime factorization method. Let us solve some of the examples here:

$ 4 = 2 \times 2 \Rightarrow \sqrt 4 =2$

$ 9 = 3 \times 3 \Rightarrow \sqrt 9 =3$

$ 16 = 2 \times 2 \times 2 \times 2 \Rightarrow \sqrt 16 =4$

$ 169 = 13 \times 13 \Rightarrow \sqrt 169 =13$

Finding Square Roots by Repeated Subtraction Method

As per the repeated subtraction method, if a number is a perfect square, then we can determine its square root by:

Repeatedly subtracting consecutive odd numbers from it Subtract till the difference is zero Number of times we subtract is the required square root For example, let us find the square root of 16.

16-1=15

15-3=12

12-5=7

7-7=0

Since , the subtruction is done for 4 times hene the square root of 16 is 4.

Square Root by Long Division Method

Finding square roots for the imperfect numbers is a bit difficult but we can calculate using a long division method. This can be understood with the help of the example given below. Consider an example of finding the square root of 1849.

Square Root of Perfect squares

The square root of a perfect square is a number that, when multiplied by itself, produces the perfect square. For example, the square root of 16 is 4 because 4 x 4 = 16.

Finding the Square Root of a Perfect Square

There are a few different ways to find the square root of a perfect square. One way is to use a calculator. Simply enter the number into the calculator and press the “√” button.

Another way to find the square root of a perfect square is to use the following formula:

$√x = x^{1/2}$

where x is the perfect square.

For example, to find the square root of 16, we would use the following formula:

$√16 = 16^{1/2} = 4$

Examples of Square Roots of Perfect Squares

Here are a few examples of square roots of perfect squares:

• √1 = 1
• √4 = 2
• √9 = 3
• √16 = 4
• √25 = 5
• √36 = 6
• √49 = 7
• √64 = 8
• √81 = 9
• √100 = 10

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Square Root Table (1 to 50)

The square root of a number is a number that, when multiplied by itself, produces the original number. For example, the square root of 9 is 3 because 3 x 3 = 9.

The following table shows the square roots of the numbers from 1 to 50:

Square Root of Decimal

A decimal value will have a dot (.) such as 3.8, 5.2, 6.33, etc. For a whole number, we have understood how to derive the square root but let us see how to get the square root of a decimal.

Example: Find the square root of 0.09.Let $N^2 = 0.09$ Taking root on both sides.

$N = ±√0.09$

As we know,

$0.3$ x $0.3 = (0.3)^2 = 0.09$

Therefore,

$N = ±√(0.3)^2$

$N = ±(0.3)$

Square Root of Negative Number

The square root of a negative number is a complex number, which means it has both a real and an imaginary part. The real part is the same as the square root of the absolute value of the negative number, and the imaginary part is the square root of -1, which is denoted by the letter i.

For example, the square root of -9 is 3i. This is because the square root of 9 is 3, and the square root of -1 is i.

Here are some other examples of square roots of negative numbers:

• The square root of -16 is 4i.
• The square root of -25 is 5i.
• The square root of -36 is 6i.

Square roots of negative numbers are used in many applications, such as electrical engineering, quantum mechanics, and signal processing.

Square root of Complex Numbers

The square root of a complex number is a complex number that, when multiplied by itself, produces the original complex number. For example, the square root of $4 + 3i$ is $2 + i$, since $(2 + i)(2 + i) = 4 + 3i$.

To find the square root of a complex number, we can use the following formula:

$\sqrt {a + bi} = \pm \left(\sqrt {\frac{ \sqrt {a^2 + b^2 + a }}{2}} + i\sqrt {\frac{ \sqrt {a^2 + b^2 - a }}{2}}\right)$

where a and b are the real and imaginary parts of the complex number, respectively.

Here are some additional examples of square roots of complex numbers:

• $ \sqrt {1 + i} = \pm \left(\sqrt {\frac{\sqrt 3}{2}} + \sqrt {\frac{\sqrt1}{2}}i \right)$
• $\sqrt {-1} = i$

How to Solve the Square Root Equation?

A square root equation is an equation that contains a square root term. The most common type of square root equation is the quadratic equation, which has the form:

$ax^2 + bx + c = 0$

where $a, b,$ and $c$ are constants and x is the variable.

To solve a quadratic equation, you can use the quadratic formula:

$x = (-b ± √(b^2 - 4ac)) / 2a$

where $√(b^2 - 4ac)$ is the square root of the discriminant.

Example:

Solve the quadratic equation:

$x^2 - 4x - 5 = 0$

Using the quadratic formula, we have:

$x = (-(-4) ± √((-4)^2 - 4(1)(-5))) / 2(1)$

$x = (4 ± √(16 + 20)) / 2$

$x = (4 ± √36) / 2$

$x = (4 ± 6) / 2$

$x = 5$ or $x = -1$

Therefore, the solutions to the quadratic equation $x^2 - 4x - 5 = 0$ are $x = 5$ and $x = -1.$

Squaring a Number

Squaring a number is the process of multiplying a number by itself. For example, the square of 5 is 5 * 5 = 25.

Squaring a number can be done using a variety of methods, including:

• Using a calculator. This is the most straightforward way to square a number. Simply enter the number into the calculator and press the “$x^2$” button.
• Using the multiplication table. You can also square a number by using the multiplication table. For example, to square 5, you would find the row for 5 in the multiplication table and then multiply 5 by itself.
• Using mental math. If you are good at mental math, you can also square a number in your head. To do this, you would first multiply the number by itself. Then, you would add the original number to the product. For example, to square 5, you would first multiply 5 by itself to get 25. Then, you would add 5 to 25 to get 30.

Examples of Squaring Numbers

Here are some examples of squaring numbers:

• $2^2 = 4 $
• $3^2 = 9$
• $4^2 = 16$
• $5^2 = 25$
• $6^2 = 36$
• $7^2 = 49$
• $8^2 = 64$
• $9^2 = 81$
• $10^2 = 100$

Applications of Squaring Numbers

Squaring numbers has a variety of applications in mathematics and science. For example, squaring numbers is used to:

• Find the area of a square.
• Find the volume of a cube.
• Calculate the distance between two points.
• Solve quadratic equations.
• Graph quadratic functions.

Squaring numbers is a fundamental operation in mathematics that has a wide range of applications.

List of Square Roots of Numbers

The square root of a number is a number that, when multiplied by itself, produces the original number. For example, the square root of 9 is 3 because 3 x 3 = 9.

The square roots of numbers can be found using a variety of methods, including:

• The Babylonian method: This is an ancient method that uses successive approximations to find the square root of a number.
• The Newton-Raphson method: This is a more modern method that uses calculus to find the square root of a number.
• The calculator: Most calculators have a built-in function for finding the square root of a number.

The following table lists the square roots of the first 100 numbers:

Examples of Square Roots in Real Life

The square root of a number has many applications in real life. For example:

• The square root of 2 is used to calculate the length of the diagonal of a square.
• The square root of 3 is used to calculate the volume of a cube.
• The square root of 5 is used to calculate the golden ratio, which is a special number that has many applications in art and design.

The square root of a number is a powerful tool that can be used to solve a variety of problems. By understanding the concept of the square root, you can open up a whole new world of mathematical possibilities.

Solved Examples on Square Roots

Example 1: Finding the Square Root of a Perfect Square

Find the square root of 144.

Solution:

Since 144 is a perfect square (i.e., it can be expressed as the square of an integer), we can simply find its square root by taking the square root of the number.

√144 = 12

Therefore, the square root of 144 is 12.

Example 2: Finding the Square Root of a Negative Number

Find the square root of -9.

Solution:

The square root of a negative number is not a real number. Instead, it is an imaginary number, which is a number that can be expressed as the product of a real number and the imaginary unit i, where i = √-1.

The square root of -9 is therefore √-9 = 3i.

Practice Questions on Square roots

Practice Questions on Square Roots

1. Find the square root of 144.

Answer: 12

Explanation: To find the square root of 144, we can use the prime factorization method. 144 = 12 x 12, so the square root of 144 is 12.

2. Find the square root of 256.

Answer: 16

Explanation: To find the square root of 256, we can use the prime factorization method. 256 = 16 x 16, so the square root of 256 is 16.

3. Find the square root of 400.

Answer: 20

Explanation: To find the square root of 400, we can use the prime factorization method. 400 = 20 x 20, so the square root of 400 is 20.

4. Find the square root of 625.

Answer: 25

Explanation: To find the square root of 625, we can use the prime factorization method. 625 = 25 x 25, so the square root of 625 is 25.

5. Find the square root of 900.

Answer: 30

Explanation: To find the square root of 900, we can use the prime factorization method. 900 = 30 x 30, so the square root of 900 is 30.

6. Find the square root of 1296.

Answer: 36

Explanation: To find the square root of 1296, we can use the prime factorization method. 1296 = 36 x 36, so the square root of 1296 is 36.

7. Find the square root of 1600.

Answer: 40

Explanation: To find the square root of 1600, we can use the prime factorization method. 1600 = 40 x 40, so the square root of 1600 is 40.

8. Find the square root of 1936.

Answer: 44

Explanation: To find the square root of 1936, we can use the prime factorization method. 1936 = 44 x 44, so the square root of 1936 is 44.

9. Find the square root of 2304.

Answer: 48

Explanation: To find the square root of 2304, we can use the prime factorization method. 2304 = 48 x 48, so the square root of 2304 is 48.

10. Find the square root of 2704.

Answer: 52

Explanation: To find the square root of 2704, we can use the prime factorization method. 2704 = 52 x 52, so the square root of 2704 is 52.

Frequently Asked Questions (FAQs) on Square root

What is a square root in Maths?

Square Root in Maths

The square root of any number is equal to a number, which when squared gives the original number. Let us say m is a positive integer, such that $√(m.m) = √(m^2) = m$

In mathematics, a square root function is defined as a one-to-one function that takes a positive number as an input and returns the square root of the given input number.

$f(x) = √x$

Examples of Square Roots

Here are some examples of square roots:

• √4 = 2
• √9 = 3
• √16 = 4
• √25 = 5
• √36 = 6
• √49 = 7
• √64 = 8
• √81 = 9
• √100 = 10

Properties of Square Roots

The following are some properties of square roots:

• The square root of a positive number is always positive.
• The square root of a negative number is always imaginary.
• The square root of 0 is 0.
• The square root of 1 is 1.
• The square root of a product of two numbers is equal to the product of the square roots of those numbers. For example, √(9 x 16) = √9 x √16 = 3 x 4 = 12.
• The square root of a quotient of two numbers is equal to the quotient of the square roots of those numbers. For example, √(9 / 16) = √9 / √16 = 3 / 4 = 0.75.

How to find square root?

How to Find the Square Root

The square root of a number is a number that, when multiplied by itself, produces the original number. For example, the square root of 9 is 3 because 3 x 3 = 9.

There are a few different ways to find the square root of a number. One way is to use a calculator. Simply enter the number into the calculator and then press the square root button.

Another way to find the square root of a number is to use the following formula:

$√x = x^{1/2}$

where x is the number you want to find the square root of.

For example, to find the square root of 9, you would use the following formula:

$√9 = 9^{1/2} = 3$

What is the meaning of this symbol ‘√’?

The symbol ‘√’ is the mathematical symbol for the square root. It is used to indicate the positive square root of a number. For example, the square root of 9 is 3, which can be written as √9 = 3.

The square root of a number is a number that, when multiplied by itself, produces the original number. In other words, if x is the square root of y, then $x^2 = y.$

What are squares and square roots?

Squares

In mathematics, a square is a regular quadrilateral, which means that it has four equal sides and four right angles. Squares are also parallelograms, which means that their opposite sides are parallel.

The area of a square is equal to the length of one side squared. For example, if a square has a side length of 5 units, then its area is $5^2 = 25$ square units.

The perimeter of a square is equal to the sum of the lengths of all four sides. For example, if a square has a side length of 5 units, then its perimeter is $4 \times 5 = 20$ units.

Square Roots

The square root of a number is a number that, when multiplied by itself, produces the original number. For example, the square root of 9 is 3 because $3 \times 3 = 9.$

Square roots can be positive or negative. The positive square root of a number is the number that is usually meant when the term “square root” is used. The negative square root of a number is the number that, when multiplied by itself, also produces the original number. For example, the negative square root of 9 is -3 because $-3 \times -3 = 9.$

Examples

Here are some examples of squares and square roots:

• The square of 5 is 25.
• The square root of 9 is 3.
• The square of -4 is 16.
• The square root of -9 is -3.

How to find the square root of perfect squares?

Finding the square root of a perfect square involves determining the number that, when multiplied by itself, results in the given perfect square. Here’s a step-by-step explanation with examples:

Step 1: Identify the Perfect Square

• A perfect square is a number that can be expressed as the product of two equal factors. For example, 9 is a perfect square because it can be written as 3 x 3.

Step 2: Prime Factorization

• Prime factorization involves breaking down a number into its prime factors, which are numbers that can only be divided by themselves and 1 without leaving a remainder.
• For example, the prime factorization of 36 is 2 x 2 x 3 x 3.

Step 3: Pair the Prime Factors

• Group the prime factors into pairs of equal factors.
• In the case of 36, we have two 2s and two 3s.

Step 4: Multiply the Paired Factors

• Multiply the paired factors to obtain the square root.
• For 36, the square root is √(2 x 2) x √(3 x 3) = 2 x 3 = 6.