219 LearnersLast updated on May 26th, 2025
Square Root of 65
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in various fields such as vehicle design, finance, etc. Here, we will discuss the square root of 65.
What is the Square Root of 65?
The square root is the inverse of the square of the number. 65 is not a perfect square. The square root of 65 is expressed in both radical and exponential form.
In the radical form, it is expressed as √65, whereas (65)1/2 in exponential form. √65 ≈ 8.06226, 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 65
The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where 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 65 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 65 is broken down into its prime factors:
Step 1: Finding the prime factors of 65 Breaking it down, we get 5 x 13: 51 x 131
Step 2: Now we found out the prime factors of 65. The second step is to make pairs of those prime factors. Since 65 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 65 using prime factorization alone doesn't provide an exact integer result.
Square Root of 65 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 65, we need to group it as 65.
Step 2: Now we need to find n whose square is less than or equal to 65. We can say n as ‘8’ because 8 x 8 = 64, which is less than 65. Now the quotient is 8, and after subtracting 64 from 65, the remainder is 1.
Step 3: Since the remainder is less than the divisor and we need more precision, we add a decimal point and bring down two zeros to make the new dividend 100.
Step 4: Double the quotient (8), making it 16, and find a number x such that 16x x ≤ 100. We find x as 0, so 160 x 0 = 0.
Step 5: Subtracting 0 from 100 gives us a remainder of 100.
Step 6: Bring down another set of zeros, making the new dividend 10000. Now, find x such that 1600x x ≤ 10000. We find x as 6, because 1606 x 6 = 9636.
Step 7: Subtract 9636 from 10000, leaving a remainder of 364.
Step 8: Continue this process to achieve the desired precision. The quotient builds up to approximately 8.06.
Square Root of 65 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 65 using the approximation method.
Step 1: We need to find the closest perfect squares to √65. The smallest perfect square less than 65 is 64, and the largest perfect square greater than 65 is 81. √65 falls somewhere between 8 and 9.
Step 2: Now we apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)
Using the formula, (65 - 64) / (81 - 64) ≈ 1 / 17 ≈ 0.059 Using the formula, we identified the decimal point of our square root.
The next step is adding the value we got initially to the decimal number which is 8 + 0.059 = 8.059, so the square root of 65 is approximately 8.06.
Common Mistakes and How to Avoid Them in the Square Root of 65
Students do make mistakes while finding the square root, such as forgetting about the negative square root and skipping long division methods. Now let us look at a few of those mistakes that students tend to make in detail.
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number has both positive and negative square roots. However, we will be taking only the positive square root, as it is the required one.
For example, √65 ≈ 8.06, there is also -8.06 which should not be forgotten.
Mistake 2
Not adding square root symbol in proper places
Not using the square root properly is another mistake that children make. To resolve this, we need to teach children that simplifying the numbers inside the square root is important and then move on to the second step.
For example, √(9 + 16) ≠ √9 + √16. The right way of using roots is √(9 + 16) = √25 = 5.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to children 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, √65 ≈ 8 is wrong; the correct answer is √65 ≈ 8.06226.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube root and square root constantly as this will help them keep away from mistakes.
For example, √65 and ∛65 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, which results in a wrong answer at the end of the sum. This may either happen in subtracting or elsewhere.
Square root of 65 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √65?
The area of the square is approximately 65 square units.
Explanation
The area of the square = side². The side length is given as √65. Area of the square = side² = √65 x √65 = 65. Therefore, the area of the square box is approximately 65 square units.
Problem 2
A square-shaped building measuring 65 square feet is built; if each of the sides is √65, what will be the square feet of half of the building?
32.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped. Dividing 65 by 2 = we get 32.5. So half of the building measures 32.5 square feet.
Problem 3
Calculate √65 × 5.
Approximately 40.31
Explanation
The first step is to find the square root of 65, which is approximately 8.06226. The second step is to multiply 8.06226 by 5. So 8.06226 × 5 ≈ 40.31.
Problem 4
What will be the square root of (60 + 5)?
The square root is approximately 8.06.
Explanation
To find the square root, we need to find the sum of (60 + 5). 60 + 5 = 65, and then √65 ≈ 8.06. Therefore, the square root of (60 + 5) is approximately ±8.06.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √65 units and the width ‘w’ is 30 units.
We find the perimeter of the rectangle as approximately 76.12 units.
Explanation
Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√65 + 30) ≈ 2 × (8.06 + 30) = 2 × 38.06 ≈ 76.12 units.
FAQ on Square Root of 65
1.What is √65 in its simplest form?
2.Mention the factors of 65.
3.Calculate the square of 65.
4.Is 65 a prime number?
5.65 is divisible by?
Important Glossaries for the Square Root of 65
- Square root: A square root is the inverse of a square. Example: 42 = 16, and the inverse of the square is the square root that is √16 = 4.
- Irrational number: An irrational number is a number that cannot be written in the form of p/q, q is not equal to zero, and p and q are integers.
- Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. This is known as the principal square root.
- Prime number: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
- Perfect square: A perfect square is a number that is the square of an integer. For example, 64 is a perfect square because it is 8 squared (8 x 8).
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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.
