Square root of 176
2026-02-28 09:57 Diff
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Last updated on August 5, 2025

Let’s discuss what is a square root? The square root of a number is a number, when multiplied by itself gives the number. Radical symbol (√) is the symbol used to indicate square root. The square root of 25 is ±5. We use it in our daily life in physics, engineering, finance etc.

What is the square root of 176

In this article, the square root of 176 is ±13.26. The square root of 176 is expressed as √176, in radical form. It is expressed as (176)½ in exponential form.
 

Finding the square root of 176

If a student wants to find the square root of a number. Which are the methods they can use?  Some common methods are,

  • Prime factorization
  • Approximation
  • Subtraction method

Let’s check the square root of 176 using these methods 

Square root of 176 by prime factorization

In this method, all the prime factors of the number are listed down and then from there one number from each pair is listed down. Then the numbers are multiplied together. 

By using prime factorization, let’s check out the square root of 176.

Step 1: Listing out the prime factors of 176 and pairing the same numbers.

Prime factors of 176 = 2 × 2 ×  2 × 2 × 11 


Pairing the factors, i.e, (2 × 2) × (2 × 2) × 11

Step 2: multiplying a number from each pair

2 × 2 ×√11 = 4√11


Hence, the √176 is 4√11
 

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Square root of 176 by long division

Here, the given number is divided into smaller numbers to find the square root of the number.

Using the long division method, let’s check the square root of 176


Step 1: The number given, 176 has three digits so let’s pair that number. Here it is 1 and 75.


Finding the number whose square is less than or equal to the 1

 Here the number is, because 12 = 1

Step 2: So, let’s divide 1 by 1


Step 3: Now, the quotient and divisor is 1, for the new divisor we add the divisor with itself.


Step 4: Continue the process till the remainder is 0

The quotient will be the square root of the number.

Therefore, the square root of 176 is ±13.266
 

Square root of 176 by approximation

Approximation method is mainly used for not a perfect square. As 176 is not a perfect square, the approximation method can be used to find the square root of 176. 

Step 1: Find any two perfect squares numbers between which 176 lies. The two perfect squares are 132 (169)and 142 (196)

Therefore, the square root of 176 lies between 13 and 14, i.e, 13 < √176 <14

Step 2: for checking out the decimal part, the formula is 

Hence,176 - 169/  196 - 169  = 7/27 = 0.259

So, the approximate square of 176 is ±13.259

Square root of 176 by subtraction method

In the subtraction method, based on the fact, the sum of the first n odd number is n2. So, here the given number is subtracted with the odd number starting with 1. The process will go on till the given number becomes 0. The method is mainly used for perfect squares; it is not used to find the square root of 176.

Common errors and how to avoid them in the square root of 176

Students tend to make errors while finding the square root of a number. So, let’s check some common errors while finding the square root and the ways to avoid it 
 

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Problem 1

If the value of x in x² = 176

Okay, lets begin

X2 = 176


In other words, x = ±√176 
 

Explanation

or we can write the actual value of 13.26

Well explained 👍

Problem 2

The length of two sides of a right triangle is 6 and √170 units. Find the length of the longest side.

Okay, lets begin

In right-angled triangle, c2 = a2 + b2


C2  = 62 + (170)2


C2  =  36+ 170


C = √206
 

Explanation

length of langest side is C = √206

Well explained 👍

Problem 3

which is smaller, √176 or √196

Okay, lets begin

The smallest is √176
 

Explanation

Because √176 is ±13.26 whereas √169 is ±14, hence, √176 is smaller than √169

Well explained 👍

Problem 4

Find the product of 4√176 The value of √176 is ±4√11

Okay, lets begin

The value of √176 is ±4√11


4 × 4√11 = 16√11

Explanation

the product of 4 × 4√11 = 16√11
 

Well explained 👍

FAQs on square root of 176

1.What goes into 176 evenly?

The number 176 is evenly divisible by 1, 2, 4, 8, 11, 16, 22, 44, 88, and 176

2.What is the greatest factor of 176?

3.Is 176 divisible by 8?

When we divide 176 by 8 is 22. So, 176 is divisible by 8. 

4.What is the prime factorization tree of 176

The prime factors of 176 is 2 × 2 × 2 × 2 × 11

176

/    \

            2     88

                   /   \

                 2     44

                        /   \

                       2      22

                              /   \

                             2      11 

Important glossaries for square root of 162

  • Quotient: The result we got after dividing two numbers is the quotient
  • Radical symbol: The radical symbol (√) is the symbol used to denote square root.
  • Square of a number: The Square of a number is the product of the number when multiplied by itself.
  • Prime factorization: It is the process of breaking down a number into prime numbers. 

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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.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.