Square root of 77
2026-02-28 08:56 Diff
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Last updated on August 5, 2025

When someone asks you to explain a square root, you can just tell that it is a number when multiplied by itself produces the same number. As we continue with our explanation, let’s assume the value of 77 Here 77 is considered as a non-perfect square root since it contain either decimal or fraction. Let's learn more about square roots in this article.

What is the square root of 77?

The square root of 77 can be easily found out by using long division method. In which it is discovered that the cumulative approximation of √77 is 8.775.

Finding the square root of 77.

There are many ways through which students can find square roots, and some of these methods are very popular. Some of the methods have been explained in detail below.
 

Square root of 77 using the prime factorization method.

In this method, we decompose the number into its prime factors.


Prime factorization of 77: 77 = 11 × 7


Since not all prime factors can be paired, 77 cannot be simplified into a perfect square. Therefore, the square root of 77 cannot be expressed in a simple radical form.

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Square root of 77 using the division method.

For non-perfect squares, we often use the nearest perfect square to estimate the square root. Follow these steps:


Step 1: Write the number 77 to perform long division.


Step 2: Identify a perfect square number that is less than or equal to 77. For 77, that number is 64 (8²)

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Step 3: Divide 77 by 8. The remainder will be 13, and the quotient will be 8.


Step 4: Bring down the remainder (13) and append two zeros. Add a decimal point to the quotient, making it 8.0.


Step 5: Double the quotient to use as the new divisor, which gives 16.


Step 6: Select a number that, when multiplied by the new divisor, results in a product less than or equal to 1300.


Step 7: Continue the division process to find √77 to the desired decimal places. → √77 ≈ 8.775.
 

Square root of 77 using the approximation method

In the approximation method, we estimate the square root by identifying the closest perfect squares surrounding the number.


Step 1: The nearest perfect squares to 77 are √64 = 8 and √81 = 9.


Step 2: Since 77 is between 64 and 81, we know the square root will be between 8 and 9.


Step 3: By testing numbers like 8.7, 8.8, and further, we find that √77 ≈ 8.775.
 

Common Mistakes and How to Avoid Them in the Square Root of 77

Here are some common mistakes students should avoid while learning to calculate the square root of 77.
 

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Problem 1

Simplify √32 ÷ √2.

Okay, lets begin

= √32 ÷ √2


= √16


= 4
 

Explanation

 Using the property of the division method, we get √16, and the square root of 16 is ±4.
 

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Problem 2

Find the square root of 36.

Okay, lets begin

√36= 6
 

Explanation

The square root of 36 is going to be 6 as 6 multiplied by itself leads to 36 
 

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Problem 3

Simplify √7 × √49

Okay, lets begin

√7 × √49


= √343
 

Explanation

Here, we multiply 7 by 49 to get 343, which results in √343. Since 343 is not a perfect square, the result cannot be simplified further.
 

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FAQs on the square root of 77

1.What is the square root of 121?

By applying the long division method on 121 we get to know that 11 divides 121 to 0 using 11 meaning 11 × 11 is equal to 121, which makes 11 the square root of 121.
 

2.How do you simplify 3√72?

 3√72 can be simplified to 18√2, as we can express √72 as 6√2. Therefore, 3 × 6 is equal to 18 hence it will be written as 18√2.
 

3.Is 77 a prime number?

 No, If we use long division on 77 we get to know that it has divisors more than just 1 and itself, so it is not a prime number. It also has its own prime factors.
 

4.What is the prime factorization of 77?

Using the prime factorization method we can easily find out that 76 can be written as multiples of 2 and 19 to be more specific 77=7×11.
 

5.What is the difference between square root and cube root?

To find out what number 4 is the square root of we need to multiply the number 4 with itself, the resulting number would be the answer in this case 4 × 4 is equal to 16.
 

Important Glossaries for Square Root of 77

  • Square Root: A number which when is multiplied by itself gives the original number is called a square root.
  • Perfect Square: A number that is the integral square of an integer I such that n = I², example I = 1, 2, 3, n = 1, 4, 9, 16, etc.
  • Prime Factorization: The ability to factorize a number in to the product of the basic arithmetic numbers, also known as primary numbers.
  • Non-Perfect Square: A figure that cannot be converted into an integer figure once divided by itself (e.g., 76).
  • Approximation Method: Approximating square root, that is, finding the closest integer which, when squared, yields the number being approximated.

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