Square Root of 77
The square root of 77 is expressed as √77 in the radical form and as (77)^{½} or (77)^{0.5} in the exponent form. The square root of 77 rounded up to 7 decimal places is 8.7749644. It is the positive solution of the equation x^{2} = 77.
 Square Root of 77: 8.774964387392123
 Square Root of 77 in exponential form: (77)^{½} or (77)^{0.5}
 Square Root of 77 in radical form: √77
1.  What Is the Square Root of 77? 
2.  Is Square Root of 77 Rational or Irrational? 
3.  How to Find the Square Root of 77? 
4.  Challenging Questions 
5.  FAQs on Square Root of 77 
What Is the Square Root of 77?
The square root of a number n is written as √n. This number when squared or multiplied by itself results in the original number n. The square root of 77 can be represented in the following ways.
 Radical form: √77
 Decimal form: 8.77496
 Exponent form: (77)^{½}
Is Square Root of 77 Rational or Irrational?
 77 is not a perfect square, which means that it does not have a natural number as its square root.
 Square root of 77 in the decimal form is √77 = 8.77496
 Square root of 77 cannot be expressed as a fraction of the form p/q. This indicates that the square root of 77 is an irrational number.
How to Find the Square Root of 77?
There are two ways to find the square root of 77:
 Long Division Method
 Prime Factorization Method
Long Division Method
The square root of 77 by long division method consists of the following steps:
 Step 1: Starting from the right, we will pair up the digits 77 by placing a bar above 77. We will also pair the 0s in the decimal from left to right.
 Step 2: Find a number that, when multiplied to itself, gives a product less than or equal to 77. The number 8 fits here as 8 square gives 64. Dividing 77 by 8 with quotient as 8, we get the remainder as 13.
 Step 3: Drag a pair of 0’s down and place them next to 13 to make the dividend 1300.
 Step 4: Double the divisor 8, and enter 16 below with a blank digit on its right. Think of a number which is greater than or equal to the dividend, i.e., 1300. 167 is the perfect number to divide 1300.
 Step 5: Multiply 167 by 7 (167 × 7 = 1169 < 1300) and write the remainder, i.e. 131.
 Step 6: Repeat this process until you get the desired number.
Therefore, the square root of 77 = 8.77
Prime Factorization Method
 To find the square root of 77, we will first express it in terms of its prime factors.
Prime factorization of 77 = 7 × 11  Since all the prime factors of 77 are unique, none of these factors are perfect squares. The square root of 77 cannot be simplified.
 Prime factorization of 77 = 7 × 11
Square root of 7 is √7 = 2.645
Square root of 11 is √11 = 3.3166  Therefore, the square root of √77 = 8.77
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Challenging Questions
 What is the negative root of 7700?
 Find the square root of 7700 up to 5 decimal places.
 What is the square root of:
a) 770
b) 777
Solved Examples

Example 1: Jack is multiplying a number by itself. If the product is 77, help Jack find the number.
Solution:
To find the number, let us assume the number to be z
On multiplying z times z = z × z = 77
z² = 77
z = √77
z = 8.77
(8.77 × 8.77 = 76.9129 ≅ 77)
The number is 8.77. 
Example 2: The area of a squareshaped table is 5929 square units. Calculate the length of one side of the table.
Solution:
The area of the table = 5929 sq. units
To find the side of a squareshaped table, let us take square root of 5929 by prime factorization method.
√5929 = √(7 × 7 × 11 × 11) = 7 × 11
Square root of 5929 is 77.
Therefore, the length of the side of the square table is 77 units. 
Example 3: Help Joy find the square root of 77 up to 3 decimal places.
Solution:
 Step 1: Starting from the right, we will pair up the digits 77 by placing a bar above 77. We will also pair the six 0s in decimals in pairs of 2 from left to right.
 Step 2: Find a number that, when multiplied to itself, gives a product less than or equal to 77. The number 8 fits here as 8 square gives 64. Dividing 77 by 8 with quotient as 8, we get the remainder as 13.
 Step 3: Drag a pair of 0’s down and place them next to 13 to make the dividend 1300.
 Step 4: Double the divisor 8 and place 16 below with a blank digit to its right. Think of a number which is greater than or equal to the dividend, i.e. 1300. 167 is the perfect number to divide 1300.
 Step 5: Multiply 167 by 7 (167 × 7 = 1169 < 1300) and write the remainder, i.e. 131.
 Step 6: Repeat this process up to three decimal places.
FAQs on the Square Root of 77
What is the Value of the Square Root of 77?
The square root of 77 is 8.77496.
Why is the Square Root of 77 an Irrational Number?
Upon prime factorizing 77 i.e. 7^{1} × 11^{1}, 7 is in odd power. Therefore, the square root of 77 is irrational.
If the Square Root of 77 is 8.775. Find the Value of the Square Root of 0.77.
Let us represent √0.77 in p/q form i.e. √(77/100) = 0.77/10 = 0.877. Hence, the value of √0.77 = 0.877
Evaluate 16 plus 9 square root 77
The given expression is 16 + 9 √77. We know that the square root of 77 is 8.775. Therefore, 16 + 9 √77 = 16 + 9 × 8.775 = 16 + 78.975 = 94.975
What is the Square of the Square Root of 77?
The square of the square root of 77 is the number 77 itself i.e. (√77)^{2} = (77)^{2/2} = 77.
What is the Square Root of 77?
The square root of 77 is an imaginary number. It can be written as √77 = √1 × √77 = i √77 = 8.774i
where i = √1 and it is called the imaginary unit.
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