Square Root of 66

The square root is the inverse of the square of the number. 66 is not a perfect square. The square root of 66 is expressed in both radical and exponential form.

In the radical form, it is expressed as √66, whereas in the exponential form it is (66)^1/2. √66 ≈ 8.12404, 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.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the prime factorization method is not used; instead, the long division method and approximation method are used. Let us now learn the following methods: -

1. Prime factorization method 
2. Long division method 
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 66 is broken down into its prime factors.

Step 1: Finding the prime factors of 66 Breaking it down, we get 2 x 3 x 11: 2^1 x 3^1 x 11^1

Step 2: Now we found out the prime factors of 66. The second step is to make pairs of those prime factors. Since 66 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 66 using prime factorization is not straightforward.

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 66, we consider it as 66.

Step 2: Now we need to find n whose square is less than or equal to 66. We can say n is ‘8’ because 8 x 8 = 64, which is less than or equal to 66. Now the quotient is 8; after subtracting 64 from 66, the remainder is 2.

Step 3: Bring down two zeros to make the new dividend 200.

Step 4: Double the quotient and add it to the current divisor. Thus, we get 8 x 2 = 16, which will be our new divisor.

Step 5: Find the largest digit x such that 16x multiplied by x is less than or equal to 200. We find x is 1.

Step 6: Subtract 161 from 200 to get a remainder of 39.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 3900.

Step 8: Repeat the process to get more decimal places in the quotient until the desired precision is reached.

So, the square root of √66 is approximately 8.124.

The approximation method is another method for finding the 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 66 using the approximation method.

Step 1: We have to find the closest perfect squares around √66. The smallest perfect square less than 66 is 64, and the largest perfect square greater than 66 is 81. √66 falls somewhere between 8 and 9.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula: (66 - 64) / (81 - 64) = 2/17 ≈ 0.118. 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 + 0.118 ≈ 8.12. Thus, the square root of 66 is approximately 8.12.

• 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, where 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 why it is known as the principal square root.

• Prime factorization: Expressing a number as a product of its prime factors. For example, the prime factorization of 66 is 2 × 3 × 11.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.