Last updated on May 26th, 2025
If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The square root is used in various fields such as architecture, finance, etc. Here, we will discuss the square root of 68.
The square root is the inverse of the square of a number. 68 is not a perfect square. The square root of 68 is expressed in both radical and exponential form.
In radical form, it is expressed as √68, whereas in exponential form it is expressed as (68)1/2. √68 ≈ 8.24621, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the long division method and approximation method are used. Let us now learn the following methods:
The product of prime factors is the prime factorization of a number. Now let us look at how 68 is broken down into its prime factors.
Step 1: Finding the prime factors of 68 Breaking it down, we get 2 x 2 x 17: 22 x 171
Step 2: Now we found out the prime factors of 68. The second step is to make pairs of those prime factors. Since 68 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 68 using prime factorization directly to find a simple square root is not feasible.
The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.
Step 1: To begin with, we need to group the numbers from right to left. In the case of 68, we consider it as a whole.
Step 2: Now we need to find n whose square is less than or equal to 68. We can say n as ‘8’ because 8 x 8 = 64, which is less than 68. Now the quotient is 8 after subtracting 64 from 68, the remainder is 4.
Step 3: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 400.
Step 4: Double the quotient (8) to get the new divisor's starting digits, which gives us 16_. We need to find a digit d such that 16d x d ≤ 400.
Step 5: By trial, 162 x 2 = 324, which is less than 400.
Step 6: Subtract 324 from 400 to get 76, and bring down two zeros to make it 7600.
Step 7: Repeat the process to find the next digit, continuing until the desired precision is reached. So the square root of √68 ≈ 8.246
The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 68 using the approximation method.
Step 1: Now we have to find the closest perfect squares of √68. The smallest perfect square less than 68 is 64, and the largest perfect square more than 68 is 81. √68 falls somewhere between 8 and 9.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).
Using the formula: (68 - 64) / (81 - 64) = 4/17 ≈ 0.235 Using the formula, we identified the decimal point of our square root.
The next step is adding the value we got initially to the smaller perfect square root number which is 8 + 0.235 = 8.235, so the approximate square root of 68 is 8.235.
Students often make mistakes while finding square roots, such as forgetting about negative square roots, skipping long division steps, etc. Now let us look at a few of these mistakes in detail.
Can you help Max find the area of a square box if its side length is given as √68?
The area of the square is 68 square units.
The area of the square = side2.
The side length is given as √68.
Area of the square = side2 = √68 x √68 = 68.
Therefore, the area of the square box is 68 square units.
A square-shaped garden measures 68 square feet, if each of the sides is √68, what will be the square feet of half of the garden?
34 square feet
We can simply divide the given area by 2 as the garden is square-shaped.
Dividing 68 by 2 = 34.
So half of the garden measures 34 square feet.
Calculate √68 x 3.
Approximately 24.73863
The first step is to find the square root of 68 which is approximately 8.246.
The second step is to multiply 8.246 by 3. So 8.246 x 3 ≈ 24.73863.
What will be the square root of (64 + 4)?
The square root is 8.
To find the square root, we need to find the sum of (64 + 4). 64 + 4 = 68, and then √68 = ±8.246.
However, for practical purposes, we often take the positive root, which is approximately 8.246.
Find the perimeter of a rectangle if its length ‘l’ is √68 units and the width ‘w’ is 10 units.
The perimeter of the rectangle is approximately 36.49242 units.
Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√68 + 10) = 2 × (8.246 + 10) = 2 × 18.246 = approximately 36.49242 units.
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.
: He loves to play the quiz with kids through algebra to make kids love it.