332 LearnersLast updated on May 26th, 2025
Square Root of 9
The square root of 9 is a value “y” such that when “y” is multiplied by itself → y × y, the result is 9. The number 9 has a unique non-negative square root, called the principal square root. Square root concept are applied in real life in the field of engineering, GPS and distance calculations, for scaling objects proportionally, etc.
What Is the Square Root of 9?
The square root of 9 is ±3, where 3 is the positive solution of the equation x2 = 9. Finding the square root is just the inverse of squaring a number and hence, squaring 3 will result in 9. The square root of 9 is written as √9 in radical form, where the ‘√’ sign is called the “radical” sign. In exponential form, it is written as (9)1/2
Finding the Square Root of 9
We can find the square root of 9 through various methods. They are:
- Prime factorization method
- Long division method
- Repeated subtraction method
Square Root of 9 By Prime Factorization Method
The prime factorization of 9 can be found by dividing the number by prime numbers and continuing to divide the quotients until they can’t be separated anymore.
- Find the prime factors of 9.
- After factorizing 9, make pairs out of the factors to get the square root.
- If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs.
So, Prime factorization of 9 = 3 ×3
Square root of 9= √[3 × 3] = 3
Square Root of 9 By Long Division Method
This method is used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.
Follow the steps to calculate the square root of 9:
Step 1: Write the number 9 and draw a bar above the pair of digits from right to left.
Step 2: Now, find the greatest number whose square is less than or equal to 9. Here, it is 3 because 32=9
Step 3: now divide 9 by 3 (the number we got from Step 2) such that we get 3 as a quotient, and we get a remainder 0.
Step 4: The quotient obtained is the square root of 9. In this case, it is 3.
Square Root of 9 By Subtraction Method
We know that the sum of the first n odd numbers is n2. We will use this fact to find square roots through the repeated subtraction method. Furthermore, we just have to subtract consecutive odd numbers from the given number, starting from 1. The square root of the given number will be the count of the number of steps required to obtain 0. Here are the steps:
Step 1: take the number 9 and then subtract the first odd number from it. Here, in this case, it is 9-1=8
Step 2: we have to subtract the next odd number from the obtained number until it comes zero as a result. Now take the obtained number (from Step 1), i.e., 8, and again subtract the next odd number after 1, which is 3, → 8-3=5.
Step 3: now we have to count the number of subtraction steps it takes to yield 0 finally. Here, in this case, it takes 3 steps.
So, the square root is equal to the count, i.e., the square root of 9 is ±3.
Common Mistakes and How to Avoid Them in the Square Root of 9
When we find the square root of 4, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.
Mistake 1
Incorrectly applying the square root property
Square root do not distribute over additions. For example, √9 ≠ √4+√5
Mistake 2
Incorrectly applying square root to negative numbers, thinking √-9 has a real solution
√-9 has no real solutions, it has solutions in complex imaginary numbers.
Mistake 3
Misinterpreting the square root symbol
√9 refers to the positive principal square root. ±√9 will yield both +3 and -3.
Mistake 4
Incorrect formula for finding square roots through different methods
The formula should be applied accurately for iterative methods
Mistake 5
Assuming √9 has only one solution
√9 has two values → +3 and -3
Square Root of 9 Examples
Problem 1
Find √(4×9)×√(4×9)×√(4×9) ?
√(4×9)×√(4×9)×√(4×9)
= (4×9)×√(4×9)
= (4×9)×2×3
=216
Answer : 216
Explanation
We found the values of the square roots of 4 and 9, then simplified the expression using square root concepts and multiplied the values.
Problem 2
What is √9 multiplied by 45 and then divided by 5 ?
√9 × 45/5
= 3×9
= 27
Answer: 27
Explanation
finding the value of √9 and multiplying by 45/5.
Problem 3
A circular dish of radius equals to 3 meters. Find its area.
radius = 3 m
Area of a circle = π×(radius)2
⇒ Area of a circle = 3.14×(3)2
⇒ Area of a circle=3.14×9 = 28.26 sq. meters
Answer: The area of a circular dish is 28.26 sq. meters
Explanation
Applying the formula for finding the area of a circle with a given radius.
Problem 4
Find the length of a side of a square whose area is 9 cm²
Given, the area = 9 cm2
We know that, (side of a square)2 = area of square
Or, (side of a square)2 = 9
Or, (side of a square)= √9
Or, the side of a square = ± 3.
But, the length of a square is a positive quantity only, so, the length of the side is 3 cm.
Answer: 3 cm
Explanation
We know that, (side of a square)2 = area of square. Here, we are given with the area of the square, so, we can easily find out its square root because its square root is the measure of the side of the square
Problem 5
Find (√9 / √49) × (√36/√64)
(√9 / √49) × (√36/√64)
=(3/7)×(6/8)
= 9/28
Answer : 9/28
Explanation
we firstly found out the values of √9, √49,√36 and √64 then solved .
FAQs on 9 Square Root
1.What is the square of 9 ?
2.What is the value of √5?
3.Is 9 a perfect square or non-perfect square?
4.Is the square root of 9 a rational or irrational number?
5.How to solve √10?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Square Root of 9?
8.How do technology and digital tools in India support learning Algebra and Square Root of 9?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for Square Root of 9
- Exponential form: An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent. Ex: 2 × 2 × 2 × 2 = 16 Or, 24 = 16, where 2 is the base, 4 is the exponent
- Prime Factorization: Expressing the given expression as a product of its factors. Ex: 48=2 × 2 × 2 × 2 × 3
- Prime Numbers: Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....
- Rational numbers and Irrational numbers: The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers.
- Perfect and non-perfect square numbers: Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :3, 8, 2
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