Square Root of 68
Square root of 68 is the number which, when multiplied by itself, gives the product as 68. In this lesson, we will calculate the square root of 68 by long division method along with finding the answers to the following questions:
 Are all even numbers perfect squares?
 Is the square root of 68 irrational?
Let us find the square root of 68.
 Square Root of 68: √68 = 8.246
 Square of 68: 68^{2} = 4624
What Is the Square Root of 68?
Square root is just an inverse operation of square. In the figure shown below, 2 is the square root of 4, but 4 is a perfect square. Does that mean that numbers that are not perfect squares cannot have a square root?
Nonsquare numbers can have a square root, but they will not be whole numbers.
The square root of 68 is represented as √68. The square root of 68 is the number which when multiplied to itself gives us the result as 68.
During ancient times, Greeks found a number that could not be written in the form of p/q, where p, q are integers and q ≠ 0.
Is the Square Root of 68 Rational or Irrational?
A number that cannot be expressed as a ratio of two integers is an irrational number. The decimal form of the irrational number will be nonterminating (i.e. it never ends) and nonrecurring (i.e. the decimal part of the number never repeats a pattern). Now let us look at the square root of 68.
√68 = 8.24621125124. Do you think the decimal part stops after 8.24621125124? No, it is neverending and you cannot find a pattern in the decimal part. Thus, the square root of 68 is an irrational number.
How to Find the Square Root of 68?
We can find the square root of 68 using various methods.
1. Repeated Subtraction
2. Prime Factorization
3. Estimation and Approximation
4. Long Division
Square Root of 68 by Prime Factorization
Let's try finding the square root of 68 using prime factorization.
Prime factorization of 68 = 2 × 2 × 17
Therefore, √68 = √(2 × 2 × 17) = 2√17
2√17 is in the lowest form and cannot be simplified further. Thus, we have expressed the square root of 68 in the radical form. Can you try and express the square root of 26 in a similar way?
Square Root of 68 by Long Division Method
The value of the square root of 68 by long division method consists of the following steps.
 Step 1: Find the largest number whose square is less than or equal to the number 68. Take this number as the divisor and the quotient (8 in this case). Divide and write the remainder.
 Step 2: In the quotient, put a decimal point after 8. Bring down two zeroes to the right of the remainder. So, the new dividend is 400.
 Step 3: Double the divisor. Now a part of the divisor is 16. Think of a number whose product is very close to 400 and that is less than or equal to 400. 2 will be the next quotient. Now we get our new divisor as 162, as 162 × 2 = 324. Do the division and get the remainder.
 Step 4: Repeat this process up to three decimal places.
Explore square roots using illustrations and interactive examples
Think Tank:
 Can you think of any perfect square number after 68?
 Since (√(68)^{2})= 68, can we say that √(68) is also a square root of 68?
Challenging Questions:
 What is the value of the square root of 682?
 Simplify ((√68)^{½})^{½}
 Determine the square root of 268.
Important Notes:
 The square root of 68 in the radical form is expressed as √68
 In exponent form, the square root of 68 is expressed as 68^{½}
 The real roots of √68 are ±8.24.
Square Root of 68 Solved Examples

Example 1: Bonnie said that the value of √68 is the same as √(68). What do you think?
Solution
A negative square root cannot have real roots.
√68 has real roots, but √(68) has only imaginary roots.
Hence, they are not the same. 
Example 2: Help Devon find the difference between the square root of 68 and the square root of 24 by the long division method.
Solution
We just learnt that the square root of 68 is 8.24.
Hence, the difference between the square root of 68 and the square root of 24 is = 8.24  4.89 = 3.35.
FAQs on Square Root of 68
What is the square root of 168 simplified?
The square root of 168 with one digit decimal accuracy is 8.2
What is the square of 68?
The square of 68 is 4624 (68^{2} = 4624).
What is the square root of 68 rounded to the nearest tenth?
The square root of 68 to the nearest tenth, means to calculate the square root of 68 where the answer should only have one number after the decimal point. Thus, the square root of 68 rounded to the nearest tenth is 8.2.
What is the value of root √68?
The value of √68 = 8.246
What is the square root of 68 in the simplest radical form?
The square root of 68 in the simplest radical form is 2√17. It cannot be simplified further.
Is the square root of 68 a rational number?
The square root of 68 is 2√17 in the simplest radical form. It cannot be simplified further.
√17 is an irrational number, therefore, the square root of 68 is an irrational number.