Square Root of 165

The square root is the inverse of the square of the number. 165 is not a perfect square. The square root of 165 is expressed in both radical and exponential form. In the radical form, it is expressed as √165, whereas (165)^(1/2) in exponential form. √165 ≈ 12.8452, 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, the prime factorization method is not used for non-perfect square numbers, where long-division and approximation methods are used. Let us now learn the following methods:

• Prime factorization method
   
• Long division method
   
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 165 is broken down into its prime factors.

Step 1: Finding the prime factors of 165 Breaking it down, we get 3 x 5 x 11: 3¹ x 5¹ x 11¹

Step 2: Now we found out the prime factors of 165. The second step is to make pairs of those prime factors. Since 165 is not a perfect square, its digits cannot be grouped in pairs.

Therefore, calculating √165 using prime factorization is impractical.

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 165, we need to group it as 65 and 1.

Step 2: Now we need to find n whose square is 1. We can say n as ‘1’ because 1 x 1 is less than or equal to 1. Now the quotient is 1. After subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 65, which is the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be 2n. We need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 65. Let us consider n as 3, now 2 x 3 x 3 = 18.

Step 6: Subtract 65 from 18; the difference is 47, and the quotient is 13.

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 4700.

Step 8: Now we need to find the new divisor that is 129 because 129 x 9 = 1161.

Step 9: Subtracting 1161 from 4700, we get the result 3539.

Step 10: Now the quotient is 12.8.

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √165 is approximately 12.84.

The approximation method is another method for finding 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 165 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √165. The smallest perfect square less than 165 is 144, and the largest perfect square greater than 165 is 169. √165 falls somewhere between 12 and 13.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using this formula, (165 - 144) ÷ (169 - 144) = 0.84. Adding this decimal to our initial estimate, 12 + 0.84 = 12.84.

So the square root of 165 is approximately 12.84.

• Square root: A square root is the inverse operation of squaring a number. Example: 4^2 = 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 usually the positive square root that is more commonly used due to its applications in the real world. This is why it is known as the principal square root.
   
• Decimals: A decimal number has a whole number part and a fractional part separated by a decimal point, for example, 7.86, 8.65, and 9.42.
   
• Prime factorization: This is the process of determining which prime numbers multiply together to make the original number.