Square Root of 67.72

The square root is the inverse of the square of a number. 67.72 is not a perfect square. The square root of 67.72 is expressed in both radical and exponential form.

In the radical form, it is expressed as √67.72, whereas (67.72)^1/2 in the exponential form. √67.72 ≈ 8.228, 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 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. However, since 67.72 is not an integer, prime factorization is not applicable in the traditional sense.

For non-perfect squares like 67.72, we focus on other methods such as the long division or approximation method.

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, we need to group the numbers from right to left. In the case of 67.72, we consider two decimal places for precision.

Step 2: Identify the largest number whose square is less than or equal to 67. The number is 8 because 8 x 8 = 64. Now, the quotient is 8 and the remainder is 3.

Step 3: Bring down 72 to make the new dividend 372. Double the quotient and bring down the next pair of digits. We get 16_ as the new divisor.

Step 4: Find the largest digit (n) such that (16n) * n ≤ 372. Let n be 2; thus, (162) x 2 = 324.

Step 5: Subtract 324 from 372 to get a remainder of 48. In the quotient, we have 8.2.

Step 6: Since we want more decimal precision, bring down pairs of zeros to continue the process.

Step 7: Continue the steps until desired precision is reached. The square root of 67.72 is approximately 8.228.

The approximation method is another method for finding square roots. It is an easy method to estimate the square root of a given number. Let's learn how to find the square root of 67.72 using the approximation method.

Step 1: Identify the closest perfect squares around 67.72. The closest perfect squares are 64 and 81, where √64 = 8 and √81 = 9.

Step 2: 67.72 is closer to 64 than it is to 81. We can estimate that √67.72 is slightly more than 8.

Step 3: Use linear approximation or interpolation to refine the estimate. For example: (67.72 - 64) / (81 - 64) = 0.219, so an additional 0.219 x (9 - 8) = 0.219 should be added to 8.

Therefore, √67.72 ≈ 8.219, but refining this further with more precise calculations gives √67.72 ≈ 8.228.

• Square root: A square root is the inverse of a square. For 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, we often consider only the positive square root, known as the principal square root.

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

• Approximation: Approximation is the process of finding a value that is close to the actual value. For example: The square root of 67.72 is approximately 8.228.