Square Root of 167

The square root is the inverse operation of squaring a number. 167 is not a perfect square. The square root of 167 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √167, whereas in the exponential form, it is expressed as (167^(1/2) .The approximate value of √167 is 12.9228, which is an irrational number because it cannot be expressed as a fraction of two integers.

For perfect square numbers, the prime factorization method is effective. However, for non-perfect squares like 167, the long division method and approximation method are more suitable. Let's explore these methods:

• Prime factorization method
   
• Long division method
   
• Approximation method

Prime factorization involves expressing a number as a product of prime numbers.

For 167, the prime factorization is straightforward since 167 is a prime number itself. Thus, it cannot be factored further.

Since 167 is not a perfect square, we cannot pair its prime factors to simplify the square root. Therefore, calculating √167 using prime factorization is not feasible.

The long division method is used for finding the square roots of non-perfect square numbers. Here’s how it works for 167:

Step 1: Group the number from right to left. In the case of 167, we group it as (1)(67).

Step 2: Find a number n whose square is less than or equal to 1. Here, n is 1 since 1^2 = 1. The quotient becomes 1, and the remainder is 0.

Step 3: Bring down the next group, 67, making the new dividend 67. Add the previous divisor (1) to itself to get 2, which is part of the new divisor.

Step 4: Consider 2n as the new divisor. We need to find n such that 2n × n ≤ 67. Trying n as 3 gives 23 × 3 = 69, which is too large. Trying n as 2 gives 22 × 2 = 44, which fits.

Step 5: Subtract 44 from 67 to get a remainder of 23.

Step 6: Since the new dividend is smaller than the divisor, add a decimal point and bring down two zeros to make it 2300.

Step 7: Find the new divisor, which becomes 249 (since the previous quotient was 12). Find n such that 249n × n ≤ 2300. Trying n as 9 gives 2499 × 9 = 2241.

Step 8: Subtract 2241 from 2300 to get a remainder of 59.

Step 9: Continue this process to get more decimal places as needed.

Thus, √167 ≈ 12.9228.

Approximation is a simpler method to estimate square roots. Follow these steps for √167:

Step 1: Identify the closest perfect squares. 144 and 169 are the nearest perfect squares to 167. √167 lies between √144 (12) and √169 (13).

Step 2: Use the formula to approximate: (Given number - smaller perfect square) / (Greater perfect square - smaller perfect square). For 167, (167 - 144) / (169 - 144) = 23 / 25 = 0.92. Step 3: Add the approximation to the smaller square root value: 12 + 0.92 = 12.92.

Therefore, √167 is approximately 12.92.

• Square root: The number that, when multiplied by itself, gives the original number. Example: √16 = 4 because 4 × 4 = 16.
   
• Irrational number: A number that cannot be expressed as a simple fraction, with a non-repeating and non-terminating decimal. Example: √2.
   
• Prime number: A natural number greater than 1 that has no positive divisors other than 1 and itself. Example: 167.
   
• Approximation: The process of finding a value that is close to but not exactly equal to a specific number.
   
• Perfect square: A number that is the square of an integer. Example: 144, as 12 × 12 = 144.