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Square Root of 65
Before we begin, let's understand the meaning of square root. The symbol of square root is √ and it is an integral part of mathematics. Once you understand the basics of finding the square root of a number, you can solve any square rootrelated problem. In this short lesson, we will learn about the square root of 65 and learn how to find it using different methods like the long division method.
Let us find the square root of 65.
 Square Root of 65: √65 = 8.062
 Square of 65: 65^{2} = 4225
What Is the Square Root of 65?
The square root of a number is the number which, when multiplied by itself, gives the original number as the product. The square root of 65 is 8.0622577483. The square root of 65 in the radical form is expressed as √65 and in the exponent form, it is expressed as 65^{1/2}. The square root of 65 rounded to 5 decimal places is 8.06225.
Is the Square Root of 65 Rational or Irrational?
Any number which cannot be expressed as p/q, where p and q are integers and q is not equal to 0, are called irrational numbers. Can √65 be expressed in such a way? In the decimal form, we can see that the decimal part of √65 is 8.0622577482986... which is nonterminating, nonrepeating, and neverending. Hence, √65 is an irrational number.
How to Find the Square Root of 65?
65 is not a perfect square. We can find the square root of 65 by the approximation method. To find the accurate value, we can use the long division method. In the approximation method, we find square numbers close to 65. We see that 64 and 81 are the perfect square numbers close to 65. The square root of 64 is 8 and the square root of 81 is 9. Therefore, the square root of 65 must lie between 8 and 9 and it must be closer to 8 as 65 is closer to 64. This method only gives us an approximate answer. To know the exact value, we can use the long division method and find a more accurate decimal value for √65.
Simplified Radical Form of Square Root of 65
√65 in the simplest radical form is √65 only. We cannot express √65 as √(a×b), where a and b are integers. Thus, √65 is already in the simplest radical form.
Square Root of 65 By Long Division
Let us follow these steps to find the square root of 65 by long division.
 Step 1: Set up the number in pairs of two digits by placing a bar above it, starting from the unit's place.
 Step 2: Find a divisor such that when we multiply it to itself, the product is <= 65. We see that 8 × 8 = 64
 Step 3: Since we do not have any other digits after 65 to carry forward, we write pairs of zeros after the decimal point (as 65 = 65.000000...). Since we have added a decimal point in the dividend, let us include a decimal point in the quotient after 8.
 Step 4: Bring the 0 pair down and our new dividend is 100. We need to find a new divisor. We double the quotient 8 × 2 = 16 and find a new digit for our units place for the new divisor, such that the product of this new divisor with that number is less than or equal to 100. Here we will place 0 as the units digit for our new divisor and also place it after the decimal point in the quotient. Our new divisor is 160.
 Step 5: We bring down the next pair of zeros and our new dividend is 10000. We need to find a new divisor like how we did in the previous step. We double the quotient 8 × 2 = 16 and find the units place of our new divisor such that the product is less than or equal to 10000. We see that 1606 × 6 = 9636 which is close to 10000. Thus, we place 6 at the units place of our new divisor and place 6 in the quotient.
Bring the next pair of zeros down. We will repeat the process to find the next decimal place. The square root of 65 up to 2 decimal places is 8.06 as shown above. Can you continue the process and find the square root of 65 to 5 decimal places?
Explore square roots using illustrations and interactive examples
Important Notes:
 The square root of 65 in the radical form is expressed as √65.
 In the exponent form, the square root of 65 is expressed as 65^{1/2}.
 The real roots of √65 are 8.0622577.
Challenging Questions:
 What is the value of √√65?
 Determine the square root of 655 by the long division method.
Square Root of 65 Solved Examples

Example 1: Jessica told her friends that the value of √65 is the same as √65. What can you say about her statement?
Solution
The square root of a negative number is an imaginary number. However, √65 is a real number. Hence, they are not the same.

Example 2: Sam wants to paint the ceiling of his bedroom. The area of the ceiling is 65 sq. feet. If the roof is in the shape of a square, what is the length of the side of the ceiling?
Solution
The area of the ceiling is 65 sq. feet
Since, area = side × side
65 = 8.06 × 8.06
√65 = 8.06
Hence, the length of the side of the ceiling will be 8.06 feet.
FAQs on Square Root of 65
What is the square root of 65?
The square root of 65 is 8.0622577.
What is the square of 65?
The square of 65 is 4225.
How do you find the square root of 65?
We can find the square root of 65 by using the prime factorization method, repeated subtraction, or the long division method.
Is the square root of 65 an irrational number?
Yes, the square root of 65 is an irrational number.
What is the square root of 65 in simplified form?
The square root of 65 in simplified form is √65.
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