Square root of 49
2026-02-28 13:56 Diff
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Last updated on August 5, 2025

The square root of 49 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y × y, the result is 49. It contains both positive and a negative root, where the positive root is called the principal square root.

What Is the Square Root of 49?


The square root of 49 is ±7. The positive value, 7 is the solution of the equation x2 = 49. As defined, the square root is just the inverse of squaring a number, so, squaring 7 will result in 49.  The square root of 49 is expressed as √49 in radical form, where the ‘√’  sign is called “radical”  sign. In exponential form, it is written as (49)1/2  
 

Finding the Square Root of 49

We can find the square root of 49 through various methods. They are:

  • Prime factorization method
  • Subtraction method

Square Root of 49 By Prime Factorization Method

The prime factorization of 49 involves breaking down a number into its factors. Divide 49 by prime numbers, and continue to divide the quotients until they can’t be separated anymore.

After factoring 49, make pairs out of the factors to get the square root. If there exists numbers which cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs

So, Prime factorization of 49 = 7 × 7     


But for 49, pairs of factor 7 are obtained.


So, it can be expressed as  √49 = √(7 × 7) = 7

 
7 is the simplest radical form of √49

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Square Root of 49 By Long Division Method

This is a method 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 49:


Step 1: Write the number 49 and draw a bar above the pair of digits from right to left. 49 is a 2-digit number, so it is already a pair.


Step 2: Now, find the greatest number whose square is less than or equal to 49. Here, it is 7 Because 72=49


Step 3: Now divide 49 by 7 (the number we got from Step 2) and we get a remainder of 0.

  Step 4: The quotient obtained is the square root. In this case, it is 7.

Square Root of 49 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 49 and then subtract the first odd number from it. Here, in this case, it is 49-1=48


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., 48, and again subtract the next odd number after 1, which is 3, → 48-3=45. Like this, we have to proceed further.


Step 3: Now we have to count the number of subtraction steps it takes to yield 0 finally. Here, in this case, it takes 7 steps.

So, the square root is equal to the count, i.e., the square root of 49 is ±7.

Common Mistakes and How to Avoid Them in the Square Root of 49

When we find the square root of 49, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.
 

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Problem 1

Simplify √36 + √49+ √36 + √49 ?

Okay, lets begin

 √36 + √49 +√36 + √49

= 6 + 7 + 6 + 7

= 26

Answer : 26

Explanation

 firstly, we found the values of the square roots of 36 and 49, then added the values.
 

Well explained 👍

Problem 2

What is √49 multiplied by 7 and then subtracting 7 from it?

Okay, lets begin

 (√49 ⤬ 7) - 7

= (7⤬7)-7

= 49-7

=42


Answer:     42 
 

Explanation

 breaking √49 into the simplest form , multiplying it by 7 and then subtracting 7 from the product .
 

Well explained 👍

Problem 3

Find the radius of a circle whose area is 49π cm².

Okay, lets begin

Given, the area of the circle = 49π cm2


Now, area = πr2 (r is the radius of the circle)


So, πr2 = 49π cm2


We get, r2 = 49 cm2


r = √49 cm


Putting the value of √49 in the above equation, 


We get, r = ±7 cm


Here we will consider the positive value of 7.


Therefore, the radius of the circle is 7 cm.


Answer: 7 cm.
 

Explanation

We know that, area of a circle = πr2 (r is the radius of the circle).According to this equation, we are getting the value of “r” as 7 cm by finding the value of the square root of 49.
 

Well explained 👍

Problem 4

In a right-angled triangle, the base is 24cm and the hypotenuse is 25cm. Find the measure of the height.

Okay, lets begin

(base)2+(height)2=(hypotenuse)2


⇒ (height)2= (hypotenuse)2-(base)2


⇒ (height)2= 252-242


⇒(height)2=625-576

⇒(height)2=49


⇒height = √49


⇒ height= 7 


Answer: 7 cm
 

Explanation

According to Pythagoras Theorem, we find the measure of the height of the triangle using √49
 

Well explained 👍

Problem 5

Find √49 / √7

Okay, lets begin

√49/√7

= √(49/7)

=(√7  ╳ √7 )/√7

= √7


Answer : √7 ≅ 2.6457
 

Explanation

we first broke √49 into √7  ╳ √7 and then divided the product by √7 .
 

Well explained 👍

FAQs on 49 Square Root

1.What is the √49 in fraction?

2.What are the two square roots of 49?

The two square roots of 49 are 7 and -7.
 

3.Is 49 a perfect square or non-perfect square?

 49 is a perfect square, since 49 =(7)2.
 

4.Is the square root of 49 a rational or irrational number?

The square root of 49 is ±7. So, 7 is a rational number, since it can be obtained by dividing two integers and can be written in the form p/q, where p and q are integers.
 

5.Is √49 an integer?

Yes, √49 =7 is an integer.
 

Important Glossaries for Square Root of 49

  • 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: 3 ⤬ 3 ⤬ 3 ⤬ 3 = 81 or, 3 4 = 81, where 3 is the base, 4 is the exponent.
  • Factorization: Expressing the given expression as a product of its factors Ex: 52=2 ⤬ 2 ⤬ 13 
  • 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 :2, 8, 18

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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