Square Root of 64

Before calculating the value of the square root of 64, let’s have a look at the definition and representation of the square root. In mathematics, the square root of a number is a value that gives the original number on multiplication by itself. For example, if p is the square root of q, it is represented as p = √q, or we can express the same equation as p² = q. Here,’√’ is the radical symbol used to denote the root of numbers. The positive number, when multiplied by itself, represents the square of the number. Thus, the square root of the square of a positive number gives the original number. In this article, you will understand how to calculate the square root of 64 in different methods with detailed explanations along with representations.

What is the Square Root of 64?

The square root of 64 is 8, i.e. √64 = 8. The radical representation of the square root of 64 is √64. Also, we know that the square of 8 is 64, i.e. 8² = 8 × 8 = 64. Thus, the square root of 64 can also be expressed as √64 = √(8)² = √(8 × 8) = 8. However, “every positive real number such as 64 owns two square roots; one is positive, and the other is negative. They are inverse with each other.” They lie on opposite sides of zero on the number line and are the same distance from zero. Besides, “a square root of a number is one of its two equal factors,” for example, 3 is a square root of 9 because (3)(3) = 9. We can also say that -3 is a square root of 9 because (‒3)(‒3) = 9. By the definition given above, the symbol √q is used to indicate the non-negative or principal square root of any non-negative real number q. In the above example, √9 = 3 shows the non-negative or principal square root of 9, whereas ‒√9 = ‒3 means the negative square root of 9. So, the square root of 64 is ±8. Although the principal square root of a positive number is only one of its two square roots, the designation “the square root” is often used to refer to the principal or non-negative square root √q. Hence, we generally represent √(64) = 8, rather than ‒8. Also, we can say that 64 is a perfect square and perfect cube number.

How to Calculate the Square Root of 64

To calculate the value of the square root of 64, we use different methods in mathematics. All these methods are explained here with detailed explanations. They are:

• Repeated subtraction Method
• Prime factorization method
• Long division method

Square Root of 64 by Repeated Subtraction Method

In this method, we should subtract the successive odd numbers till we obtain zero, starting from 64 and 1. The number of odd numbers we subtract in this process is treated as the square root of 64.

• 64 – 1 = 63
• 63 – 3 = 60
• 60 – 5 = 55
• 55 – 7 = 48
• 48 – 9 = 39
• 39 – 11 = 28
• 28 – 13 = 15
• 15 -15 = 0

Thus, starting from 64, we have subtracted 8 times to obtain 0. Therefore, the square root of 64 is 8.

Click here to get more information about the square root of a number by repeated subtraction method.

Square Root of 64 by Prime factorization

In the prime factorization method, we have to divide the number by prime numbers starting from 2. When it cannot be divided further with 2, divide it with the next prime number, i.e. 3 and continue this process till we get 1.

64 ÷ 2 = 32

Again divide 32 by 2.

32 ÷ 2 = 16

Continue dividing by 2.

16 ÷ 2 = 8

8 ÷ 2 = 4

4 ÷ 2 = 2

2 ÷ 2 = 1

Finally, we received the number 1 in the division process. So that we cannot proceed further. Hence, the prime factorisation of 64 can be written as 2 x 2 × 2 x 2 x 2 x 2 or 2⁶.

The above process can also be shown as:

Therefore, the prime factorization of 64 is 2 × 2 × 2 × 2 × 2 × 2.

Let us take these factors as pairs.

That means, 64 = (2 × 2) × (2 × 2) × (2 × 2)

Here, we got three pairs of the same numbers.

Now, take the square root on both sides of the above representation.

√64 = √(2 × 2) × (2 × 2) × (2 × 2) = 2 × 2 × 2 = 8.

Hence, the square root of 64 is equal to 8.

Square Root of 64 by Long division method

Go through the procedure given below, to learn how to find the square root of 64 by long division method.

• Write 64.
• Take the digits of the number in pairs from the right. So, the entire number is chosen as a pair.
• We need to divide 64 with a number such that the number × number gives 64 or a number lesser than that.
• Let’s try with different numbers, such as 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25, 6 × 6 = 36, 7 × 7 = 49, 8 × 8 = 64.
• Thus, the divisor should be 8 which gives the corresponding quotient as 8.
• As 8 × 8 = 64, which is exactly equal to the dividend. Hence, the remainder will be 0.
• Now we got the quotient 8, and the remainder as 0.

Therefore, the quotient is equal to the square root of the number 64, i.e. √64 = 8.

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Solved Examples

Example 1:

There are 64 students in a class. These students should be seated in an equal number of rows and columns. In how many rows and columns, the seating can be done?

Solution:

Given,

Total number of students = 64

Let y be the number of rows or columns.

So, y × y = 64 {Number of rows = Number of columns)

y. = 64

y = √64

y = 8 {y cannot be -8 since the number of rows/columns must be positive}

Hence, the students of that class must be seated in 8 rows and 8 columns.

Example 2:

Evaluate: 4 × 6 + 8 × √64 – 55.

Solution:

4 × 6 + 8 × √64 – 55

= 4 × 6 + 8 × 8 – 55

= 24 + 64 – 55

= 88 – 55

= 33

Therefore, 4 × 6 + 8 × √64 – 55 = 33.

Suppose we took √64 value as -8, then:

4 × 6 + 8 × √64 – 55

= 4 × 6 + 8 × (-8) – 55

= 24 – 64 – 55

= -95