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Square Root of 78
The square root of 78 is a number which when multiplied by itself results in 78. The square root of 78 is denoted by √78. 78 can be expressed as 78 = 2 × 3 × 13. Hence, on simplifying square root of 78 is written as √78. We will now look at how to find the value of the square root of 78 and solve some problems to help give you a better understanding.
 Square Root of 78: √78 = 8.8317608...
 Square of 78: 78^{2} = 6084
1.  What is the Square Root of 78? 
2.  Is Square Root of 78 Rational or Irrational? 
3.  How to Find the Square Root of 78? 
4.  Important Notes 
5.  FAQs on Square Root of 78 
6.  Challenging Questions 
What is the square root of 78?
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 78 can be written in multiple ways:
 Radical form: √78
 Decimal form: 8.832
 Exponent form: (78)^{1/2}
Is Square Root of 78 Rational or Irrational?
 78 is a number that is not a perfect square, meaning it does not have a natural number as its square root.
 Also, its square root cannot be expressed as a fraction of the form p/q which tells us that the square root of 78 is an irrational number.
How to Find the Square Root of 78?
There are multiple ways to find the square root of 78
 Long Division Method
 Estimation and Approximation
One can find out more about these methods by clicking here.
Long Division Method
The square root of 78 by long division method consists of the following steps:
 Step 1: Starting from the right, we will pair up the digits 78 by putting a bar above them. We also pair the 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 78. The number 8 fits here as 8 square gives 64. Dividing 78 by 8 with quotient as 8, we get the remainder as 14.
 Step 3: Drag a pair of 0’s down and fill it next to 14 to make the dividend 1400.
 Step 4: Double the divisor 8, and enter 16 below with a blank digit on its right. Guess the largest possible digit(X) to fill in the blank and the quotient for which the product of 16X × X results in a value less than or equal to 1400. This will be 8 in this case, so divide 1400 with 168 and write the remainder.
 Step 5: Repeat this process to get the decimal places you want.
Therefore, the square root of 78 = 8.832.
Estimation and Approximation
The estimation method gives us an approximate answer and is usually not accurate to more than 1 decimal place. However, it is easy to perform as can be seen under.
 Step 1: Find a perfect square that is smaller than and bigger than 78. In this case, 8 and 9 will work as their squares are 64 and 81.
 Step 2: Writing in terms of inequality 8 < √78 < 9 = 64 < 78 < 81.
 Step 3: Multiply by 100 and write in terms of square roots √6400 < √7800 < √8100.
 Step 4: Move closer to inequality √7744 < √7800 < √7921 = 88 < 10√78 < 89 = 8.8 < √78 < 8.9.
 Step 5: Taking average of upper and lower limits we get, (8.8 + 8.9)/2 = 8.85.
Therefore, we can estimate the square root of 78 ≅ 8.85.
Explore Square roots using illustrations and interactive examples
Important Notes
 The square root of 78 is an irrational number.
 The number 78 is not a perfect square.
 There will be n/2 digits in the square root of an even number with n digits.
 There will be (n+2)/2 digits in the square root of an odd number with n digits.
Challenging Questions
 What is the square root of 78 up to 5 decimal places(Use Long Division)?
 Find the square root of the following numbers.
 7878
 0.0078
Solved Examples

Example 1: What is the square root of 78?
Solution:
The square root of negative numbers is represented by imaginary numbers. So, the square root of √78 is written as √78 i, where (i = √1).

Example 2: Richard wants to buy a new wall painting. In the store, he finds a square painting that has an area of 78 sq inches. How long is each side of the painting?
Solution:
To find the side of the painting, we will have to find the square root of 78. The square root is √78 = 8.832. Hence, the side length of the painting is approximately 8.8 inches.
FAQs on Square Root of 78
What is the prime factorization of 78?
The prime factorization of 78 as 78 = 2 × 39.
Is 78 a perfect square root?
78 is not a perfect square. 78 is a natural number, but since there is no other natural number that can be squared to result in the number 78, it is NOT a perfect square.
What is the decimal value of the square root of 78?
The decimal value of the square root of 78 is 8.832.
Is the square root of 78 a rational number?
No, the square root of 78 is not a rational number since the square root of 78 is nonterminating and cannot be represented in the form of p/q.
Can we find the square root of 78 by the prime factorization method?
No, we can’t find the square root of 78 by the prime factorization method. This is because its factors are both prime numbers with power 1 and therefore we cannot simplify it further.
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