Square Root of 69.25

The square root is the inverse of the square of the number. 69.25 is not a perfect square. The square root of 69.25 is expressed in both radical and exponential form.

In the radical form, it is expressed as √69.25, whereas (69.25)^(1/2) in the exponential form. √69.25 ≈ 8.320, 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 like 69.25, methods such as 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. Since 69.25 is not a perfect square, prime factorization is not a suitable method.

Instead, we use approximation or long division methods to find the square root.

The long division method is particularly used for non-perfect square numbers. Let us learn how to find the square root using the long division method, step by step:

Step 1: Group the numbers from right to left. In the case of 69.25, consider it as 69 and 25 for the decimal part.

Step 2: Find n whose square is less than or equal to 69. We find n as 8 because 8 × 8 = 64, which is less than 69. The quotient is 8, and the remainder is 69 - 64 = 5.

Step 3: Bring down 25 (from the decimal part), making it 525. Double the quotient (8) to get 16, which will be part of the new divisor.

Step 4: Find n such that 16n × n ≤ 525. Let n be 3. Then, 163 × 3 = 489.

Step 5: Subtract 489 from 525 to get the remainder 36.

Step 6: Since the dividend is less than the divisor, continue with a decimal point and add zeroes to the dividend for precision.

Step 7: The continued steps give the square root of 69.25 as approximately 8.320.

The approximation method is an easy way to find square roots. Let's learn how to find the square root of 69.25 using this method:

Step 1: Identify the closest perfect squares. The nearest perfect squares for 69.25 are 64 and 81. √69.25 falls between 8 and 9.

Step 2: Apply the formula (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square).

Using the formula: (69.25 - 64) ÷ (81 - 64) = 5.25 ÷ 17 ≈ 0.309. Adding this decimal to the smaller whole number: 8 + 0.309 = 8.309,

so the square root of 69.25 is approximately 8.320.

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

• Irrational number: A number that cannot be expressed as a simple fraction (p/q), where p and q are integers and q ≠ 0.

• Long division method: A technique used to find the square root of non-perfect squares by systematically dividing the number.

• Radical form: A way to express the square root of a number using the radical symbol (√). Example: √69.25.

• Approximation method: A method to estimate the square root of a number when it is not a perfect square.