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. 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 ‘√9’ or 91/.2.

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

Average method

Prime Factorization method

Repeated subtraction method

Number line method

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.

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.

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

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.

Draw a number line with ‘0’ as a reference point and integers labeled at unit lengths on the 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 reference point, draw a perpendicular of unit length to the number line. The length of line joining the tip of the perpendicular to the reference point gives the value of root 2 by using 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 for a number of times, at a certain instant, the line joining the free end of the perpendicular and the reference point gives the measure of 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.

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 is 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.

FAQ (Frequently Asked Questions)

1. How to find the value of root 9?

Value of root 9 is that number which when multiplied by itself gives 9 as the product. 9 can be obtained by multiplying the positive integer 3 by itself. The product of -3 with itself is also equal to 9. So, 9 is a real number which is a perfect square number which has a positive root equal to + 3 and the negative root is equal to - 3. Since 9 is a perfect square number, square root of 9 can be found by the average method, repeated subtraction method or prime factorization method. The number line method and guess and check method can also be used to find the square root of 9.

2. What is the value of the square root of - 9?

Negative integers do not have real roots. However, they have two imaginary roots: a positive imaginary root and a negative imaginary root. Square root of -1 is equal to an imaginary number unit ‘i’. The imaginary number ‘i’ is equivalent to the number ‘1’ in case of real numbers. The square root of a negative number is found by writing the number as a product of the positive part of the number with -1. The square root of the positive part is found and is multiplied by the imaginary number ‘i’.

Using this concept, square root of - 9 can be calculated as

Square root of - 9 = √-9 = √9 x (-1) = √9 x √-1 = ± 3i