Square Root of 9 Explained with Concept and Solution

The square root of any real number is that number which when multiplied by itself gives the value of the number whose square root is to be determined. The square root of a number can be represented in the exponential form of the number to the power ½. Square roots are denoted by the symbol ‘√’. All real numbers have two values of square roots: one positive root and the other negative root. However, the magnitude of both roots remains the same. The square root of the decimal number ‘9’ is represented as  \[\sqrt{9}\] or \[9 ^{\frac{1}{2}}\]. 

How to find Square Root of 9?

The square root of 9 can be determined by several methods that include the following:

1. Average method

2. Prime Factorization method

3. Repeated subtraction method

4. Number line method

How to find Square Root of 9 by Average Method?

The average method of finding the square roots of a number is determining the square roots finding the average of two square numbers between which the number lies. 

The decimal number ‘9’ lies between the square numbers 4 and 16 whose square roots are 2 and 4. The average of 2 and 4 is 

\[ Average = \frac{2 + 4}{2} = \frac{6}{2} = 3\]

Therefore the square root of 9 found using the average method is equal to 3.

An average of a list of data is a mathematical expression for the center value of a set of data. It is defined mathematically as the ratio of the total of all the data to the number of units in the list. In statistics, the average of a particular collection of numerical data is also known as the mean. The average of 2, 3, and 4 equals (2+3+4)/3 = 9/3 = 3. As a result, the central value of 2,3, and 4 is 3. Thus, the definition of average is to determine the mean value of a set of data.

How to find Square Root of 9 by the Prime Factorization Method?

The prime factorization method of finding the square root of a number is the method in which the number whose square root is to be determined is expressed as the product of prime numbers. The identical prime numbers are grouped in pairs and the product of one element from each pair gives the square root of the number.

The value of root 9 is calculated by the prime factorization method by representing it as a product of its prime factors. The prime factors of 9 are 3, 3, and 1. So 9 can be expressed as:

9 = 3 x 3 x 1 x 1

Therefore value of root 9 = 3 x 1 = 3

A prime number is one with precisely two elements, one and the number itself. As an example, consider the number 30. We know that 30 = 5*6 is a prime number, but 6 is not one. The number 6 may be factored further as 2*3, where 2 and 3 are prime integers. As a result, the prime factorization of 30 is 2*3*5, with all factors being prime numbers.

Prime numbers are those with only two elements, one and the number itself. Prime numbers include 2, 5, 13, 19, and so on. 

How to find Square Root of 9 by the Repeated Subtraction Method?

Repeated subtraction method is a method in which the number whose square root is to be determined is subtracted repeatedly by consecutive odd numbers till the difference obtained is zero. The number of subtractions performed to get the difference as zero is the square root of the number. 

Square root of 9 is found by repeated subtraction method as follows:

9 - 1 = 8

8 - 3 = 5

5 - 5 = 0

The total number of subtractions performed to get the result as zero is 3. So the square root of 9 is 3.

The division is taught by repeated subtraction. It is the process of repeatedly subtracting the same number from a huge number until it reaches zero. It's a terrific technique to teach kids about division.

Repeated Subtraction, often known as division, is a method for subtracting an equal amount of elements from a collection. This approach involves continually subtracting the same number from a bigger number until the result is zero or less than the amount being removed.

How to find Square Root of 9 by the Number Line Method?

• Draw a number line with ‘0’ as a reference point and integers labeled at unit lengths on either side of the reference point. The numbers to the right of zero are positive integers and the numbers to the left of zero are negative integers. 

• Consider the reference point as the point ‘O’. From the first point on the right of the reference point, draw a perpendicular unit length to the number line. The length of the line joining the tip of the perpendicular to the reference point gives the value of root 2 by using the Pythagoras theorem. 

• If a perpendicular is drawn from the tip of the first perpendicular and the free end of the newly drawn perpendicular to the reference point is joined to the reference point, the length is equal to the value of root 3. 

• If the construction of perpendicular and joining its end to the reference point is continued many times, at a certain instant, the line joining the free end of the perpendicular and the reference point gives the measure of the value of root 3. If this length is measured with the compass and the arcs are cut from the reference point on either side of the number line, the two values of root 9 are obtained. i.e. + 3 and - 3. 

• A number line is used to compare integers at equal intervals on an endless line that extends on both sides, horizontally or vertically. The numbers grow as we travel to the right side of a number line; as we get to the left, the numbers drop.

• A number line is a visual depiction of numbers on a straight line drawn horizontally or vertically. When we write numbers on a number line, we can easily compare them and execute fundamental arithmetic operations on them. A number line is divided into three sections: the negative side, the zero side, and the positive side. Numbers to the left of 0 are all negative, whereas numbers to the right of 0 are all positive. As a result, on a number line, the numbers on the right are larger than the ones on the left. For instance, 3 is to the right of 1, hence 3 > 1. Take a look at the vertical and horizontal number lines shown below.

Fun Facts:

• Square roots of all perfect square numbers are positive or negative integers. 

• Square roots of all positive real numbers are real.

• Square roots of a negative real number are imaginary.

• Square roots of prime numbers are irrational numbers.

• Prime factorization and repeated subtraction methods can be used only to find the square roots of perfect square numbers.

Important Considerations

• If the number finishes with 2, 3, 7, or 8, it is not a perfect square.

• A perfect square number contains the numbers 1, 4, 5, 6, and 9.

• The square root of a perfect square number is simple to calculate since it yields a whole integer, whereas defective squares provide fractional or decimal values. We can simplify square a number by multiplying it by itself, but getting the square root of a number is a little more involved, especially with imperfect squares.

• The " sign represents the square root. This is known as a radical symbol or radix. The number underneath the radical sign or radix is known as the radicand. In this section, we will use a simple approach to calculate the square root of 9, where 9 is the radicand.