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What is the Square Root of 65? Definition, Methods and More

Find What is the Square Root of 65?

It is must to understand the meaning of the square root to find What is the Square Root of 65? The symbol for the square root is √ , and it is an integral part of mathematics. Once you know the basics of finding the square root of a number, you will solve any square root problem.

We need to define, analyze, simplify, and compute the square root of 65.

What is the Square Root of 65? – Definition

The square root of 65 mathematically is written with a sign like this √65.

This is in radical form – quantity (q) multiplied by itself equals 65.

√65 = q × q = q2

The square root of any number is – the number that, multiplied by itself, gives the original number. For example, the square root of 65 is 8.0622577483. In radical form, square root of 65  is √65, and in the exponential form, 65½. The square root of 65 is rounded to 5 decimal places 8.06225.

Is 65 a perfect square?

Number 65 is a perfect square if the square root of 65 equals an integer. By calculating, we can know if a number is a perfect square or not.

We show that 65 is not a perfect square with the calculation above.

Is it Rational or Irrational?

  • 65 is a perfect square – if the square root of 65 is a rational number
  • It is an irrational number – if it is not a perfect square
  • Since 65 is not a perfect square, it is an irrational number
  • It means that the answer to “the square root of 65?” will have an infinite number of decimal places.
  • The decimals don’t end, and you can’t convert them to an exact fraction.
  • In decimal form, we see that the decimal part of √65 is 8.0622577482986… which is repeatable and endless.
  • Therefore, √65 is an irrational number.

Different Methods to Simplify – What is the Square Root of 65?

With a Calculator

The easiest way to calculate is to use your calculator! Type 65 followed by √x to get the answer.

√65 ≈ 8.062257748

With a Computer

If you’re using a computer with Excel or Numbers, you can enter SQRT(65) into a cell of excel sheet. Here’s the result we got with 13 decimal places, called decimal form.

√65 ≈ 8.0622577482986

Approximation Method

  • In the process of approximation, find square numbers closer to 65. 64 and 81 are the perfect square numbers that are close to 65
  • The square root of 64 is 8, and of 81 is 9. Therefore, the square root of 65 must be between 8 and 9 and must be closer to 8 since 65 is closer to 64
  • This method only gives us an approximate answer. To find the exact value, we can use the long division method and find a more accurate decimal value for √65.

Simplified Radical Form

√65 in the simplest radical form is just √65. We cannot express √65 as √(a×b), where a and b are integers. Therefore, √65 is already in the simplest radical form.

What is the Square Root of 65? Long Division MethodWhat is the Square Root of 65_ Long Division Method

The following steps to be followed to find the square root of 65 by long division

Step 1: Set the number in pairs of two digits by placing a slash above it, starting from the unit’s location.

Step 2: A divisor to be found such that when we multiply it by itself, the product is <= 65. We see that 8 × 8 = 64

Step 3: Since we don’t have any other digits after 65 to report, we write pairs of zeros after the decimal point (65 = 65.000000…). Since we added a decimal point in the dividend, let’s include a decimal point in the quotient after 8.

Step 4: Lower the pair 0 to get new dividend 100. We need to find a new divisor. We double the quotient 8 × 2 = 16 and find a unique digit for our ones place for the new divisor so that the product of this new divisor with this number is less than or equal to 100. Here we will put 0 as the one-digit for our new divisor and 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 10,000. We need to find a new divisor as we did in the previous step, double the quotient 8 × 2 = 16 and find the ones place of our new divisor such that the product is less than or equal to 10000. We see that 1606 × 6 = 9636, close to 10000. So we put 6 in the ones place our new divisor and put 6 in the quotient.

As a Fraction

Only a rational number is shown as a fraction, and irrational numbers cannot.

We cannot convert it to an exact fraction since it is an irrational number. However, we can convert it to an approximate fraction rounded to the nearest hundredth.


≈ 8.1/1

[≈ 806/100]

≈ 8 3/50

With an Exponent

A fractional exponent can convert all square root calculations to a number (called the base).

√b = b½

√65 = 65½

Points to Remember:

In radical form, it is expressed as √65.

In exponential form, it is  65½

The real Square root of √65 is 8.0622577.

Challenging Questions:

What is the value of √65?

Determine What is the square root of 65 by the Long division method.

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