187 LearnersLast updated on May 26th, 2025
Square Root of 68
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.
What is 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.
Finding the Square Root of 68
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:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 68 by Prime Factorization Method
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.
Square Root of 68 by Long Division Method
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
Square Root of 68 by Approximation Method
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.
Common Mistakes and How to Avoid Them in the Square Root of 68
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.
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number does have both positive and negative square roots. However, we generally consider only the positive square root, as it is usually the required one.
For example: √50 ≈ 7.07, but there is also -7.07 which should not be forgotten.
Mistake 2
Not adding square root symbol in proper places
Not using the square root symbol properly is another mistake that students make. To resolve this, we need to teach students that simplifying the numbers inside the square root is important before moving to subsequent steps.
For example: √(12 + 24) ≠ √12 + √24. The correct method is √(12 + 24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to students is to simplify the numbers inside the square root. This is important so that students do not make mistakes while identifying the approximate value.
For example: √50 ≠ 7, the correct answer is √50 ≈ 7.071067.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be consistently taught the difference between cube roots and square roots to avoid confusion.
For example: √50 and ∛50 are different.
Mistake 5
Making mistakes in long division
Finding a square root using the long division method involves many steps. Due to this, many students might skip a step accidentally, resulting in an incorrect answer. This may occur during subtraction or elsewhere.
Square Root of 68 Examples
Problem 1
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.
Explanation
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.
Problem 2
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
Explanation
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.
Problem 3
Calculate √68 x 3.
Approximately 24.73863
Explanation
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.
Problem 4
What will be the square root of (64 + 4)?
The square root is 8.
Explanation
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.
Problem 5
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.
Explanation
Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√68 + 10) = 2 × (8.246 + 10) = 2 × 18.246 = approximately 36.49242 units.
FAQ on Square Root of 68
1.What is √68 in its simplest form?
2.Mention the factors of 68.
3.Calculate the square of 68.
4.Is 68 a prime number?
5.68 is divisible by?
Important Glossaries for the Square Root of 68
- Square root: A square root is the inverse operation of squaring a number. For example, 42 = 16, and the inverse operation is the square root: √16 = 4.
- Irrational number: An irrational number is a number that cannot be expressed as a simple fraction, i.e., in the form of p/q, where p and q are integers and q ≠ 0.
- Radical: A radical is an expression that uses the root symbol (√) to indicate the root of a number.
- Integer: An integer is a whole number that can be positive, negative, or zero, such as -5, 0, and 3.
- Decimal: A decimal is a number that includes a whole number and a fractional part separated by a decimal point, such as 7.86 or 9.42.
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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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
