Square Root of 65 + Solution With Free Steps

Square Root Of 65

This article is about the square root of an odd number 65, which is written in square root form as √65, which is equivalent to 8.062 in numeric form. Here the vital thing to be noted is that the square root of 65 is not a perfect square root.  

In this article, we will analyze and find the square root of 65 using various mathematical techniques, such as the approximation and long division methods.

What Is the Square Root Of 65?

The square root of the number 65 is 8.062.

The square root can be defined as the quantity that can be doubled to produce the square of that similar quantity. In simple words, it can be explained as:

√65 = √(8.062 x 8.062)

√65 = √(8.062)$^2$

√65 = ±8.062

The square can be canceled with the square root as it is equivalent to 1/2, therefore, obtaining 8.062. Hence 8.062 is 65’s square root. The square root generates both positive and negative integers.

How To Calculate the Square Root of 65?

You can calculate the square root of 65 using any of two vastly used techniques in mathematics; one is the Approximation technique, but the square root of 65 cannot be determined by this method, and the other is the Long Division method.

The symbol √ is interpreted as 65 raised to the power 1/2. So any number, when multiplied by itself, produces its square, and when the square root of any squared number is taken, it produces the actual number.

Let us discuss each of them to understand the concepts better.

Square Root by Long Division Method

The process of long division is one of the most common methods used to find the square roots of a given number. It is easy to comprehend and provides more reliable and accurate answers. The long division method reduces a multi-digit number to its equal parts.

Learning how to find the square root of a number is easy with the long division method. All you need are five primary operations- divide, multiply, subtract, bring down or raise, then repeat.

Following are the simple steps that must be followed to find the square root of 65 using the long division method:

Step 1

First, write the number 65 in the division symbol, as shown in figure 1.

Step 2

Starting from the right side of the number, divide the number 65 into pairs such as 00 and 65.

Step 3

Now divide the digit by a number, giving a number either 8 or less than 8. Therefore, in this case, the remainder is 1, whereas the quotient is 8.

Step 4

After this, bring down the next pair 100. Now the dividend is 100. To find the next divisor, we need to double our quotient obtained before. Doubling 8 gives 16; hence consider it as the next divisor.

Step 5

Now pair 16 with another number to make a new divisor that results in $\leq$ 100 when multiplied with the divisor. If the number is not a perfect square, add pair of zeros to the right of the number before starting division.

Step 6

Adding 0 to the divisor and multiplying 0 with 0 results in 0 $\leq$ 100. The remainder obtained is m1. Move the next pair of zeros down and repeat the same process mentioned above.

Step 7

Keep on repeating the same steps till the zero remainder is obtained or if the division process continues infinitely, solve to two decimal places.

Step 8

The resulting quotient of 8.062 is the square root of 65. Figure 1 given below shows the long division process in detail:

Square root of 65 by Long Division

Square Root by Approximation Method

The approximation method involves guessing the square root of the non-perfect square number by dividing it by the perfect square lesser or greater than that number and taking the average.

The given detailed steps must be followed to find the square root of 65 using the approximation technique.

Step 1

Consider a perfect square number 64 less than 65.

Step 2

Now divide 65 by √64.

65 ÷ 8 = 8.125

Step 3

Now take the average of 8 and 8.125. The resulting number is approximately equivalent to the square root of 65.

(8 + 8.125) ÷ 2 = 8.062

Square root of 65 Approximation Method

Important points

  • The number 65 is not a perfect square.
  • The number 65 is rational.
  • The number 65 can be split into its prime factorization.

Is Square Root of 65 a Perfect Square?

The number 65 is not a perfect square. A number is a perfect square if it splits into two equal parts or identical whole numbers. If a number is a perfect square, it is also rational.

A number expressed in p/q form is called a rational number. All the natural numbers are rational. A square root of a perfect square is a whole number; therefore, a perfect square is a rational number.

A number that is not a perfect square is irrational as it is a decimal number. As far as 65 is concerned, it is a perfect square / not a perfect square. It can be proved as below:

Factorization of 65 results in 5 x 13.

Taking the square root of the above expression gives:

= √(5 x 13)

= (5 x 13)$^{1/2}$

= 8.062

This shows that 65 is not a perfect square as it has decimal places; hence it is an irrational number.

Therefore the above discussion proves that the square root of 65 is equivalent to 8.062.

Square root of 65 Calculation

Images/mathematical drawings are created with GeoGebra.

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