Square Root of 67

The square root is the inverse operation of squaring a number. 67 is not a perfect square. The square root of 67 is expressed in both radical and exponential form.

In radical form, it is expressed as √67, whereas in exponential form it is expressed as (67)^1/2. √67 ≈ 8.18535, which is an irrational number because it cannot be expressed as a fraction of two integers.

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 67, we use methods such as the long-division method and approximation method. Let's learn these methods:

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

Prime factorization involves expressing a number as a product of its prime factors. However, for non-perfect squares like 67, prime factorization alone does not help in finding the square root.

Step 1: Find the prime factors of 67. 67 is a prime number, so it can only be expressed as 67 x 1. Since 67 is not a perfect square, we cannot form pairs of its prime factors.

Therefore, calculating √67 using prime factorization is not feasible.

The long division method is suitable for non-perfect square numbers. This method involves finding the closest perfect square and proceeding step by step as follows:

Step 1: Start by grouping the numbers from right to left. For 67, consider it as 67.

Step 2: Find a number n whose square is less than or equal to 67. Here, n is 8, since 8² = 64, which is less than 67.

Step 3: Subtract 64 from 67, leaving a remainder of 3.

Step 4: Bring down pairs of zeros to the right of the remainder to get 300.

Step 5: Double the divisor (8), now 16, and guess the next digit of the quotient.

Step 6: Find the largest digit x such that 16x * x is less than or equal to 300.

Step 7: Continue this process to find more decimal places. The square root of √67 is approximately 8.18535.

The approximation method provides an easy way to estimate the square root of a number.

Step 1: Identify the two closest perfect squares. For 67, they are 64 (8²) and 81 (9²). √67 is between 8 and 9.

Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

Step 3: (67 - 64) / (81 - 64) = 3 / 17 ≈ 0.1765.

Step 4: Add this decimal to the smaller integer, 8, yielding 8 + 0.1765 ≈ 8.18.

Thus, the square root of 67 is approximately 8.18.

• Square Root: A square root is a value that, when multiplied by itself, gives the original number. Example: 4² = 16, and √16 = 4.

• Irrational Number: An irrational number cannot be expressed as a simple fraction; its decimal form is non-repeating and non-terminating.

• Principal Square Root: The principal square root is the positive square root of a number.

• Prime Number: A prime number has only two distinct factors: 1 and the number itself.

• Approximation: An approximation is a value or number that is close to the exact value, often used for simplifying calculations.