Square Root of 69

The square root is the inverse of squaring a number. 69 is not a perfect square. The square root of 69 is expressed in both radical and exponential forms.

In radical form, it is expressed as √69, whereas in exponential form, it is written as 69^1/2.

√69 ≈ 8.30662, 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 typically used for perfect square numbers. However, for non-perfect square numbers, we use methods like long division and approximation. Let us now learn the following methods:

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

The product of prime factors gives the prime factorization of a number. Let's see how 69 is broken down into its prime factors:

Step 1: Finding the prime factors of 69 Breaking it down, we get 3 x 23. Therefore, the prime factorization of 69 is 3¹ x 23¹.

Step 2: Since 69 is not a perfect square, the prime factors cannot be paired.

Therefore, calculating the square root of 69 using prime factorization directly is not feasible.

The long division method is particularly useful for non-perfect square numbers. In this method, we check for the closest perfect square number to the given number. Here is how to find the square root using the long division method, step by step:

Step 1: To begin with, we need to group the digits of 69 starting from the right. In this case, we group it as 69.

Step 2: Find a number whose square is less than or equal to the leftmost group. Since 8² = 64 is less than 69, the initial quotient is 8.

Step 3: Subtract 64 from 69, resulting in a remainder of 5. Step 4: Bring down pairs of zeros one at a time. The new dividend is 500.

Step 5: Double the quotient and use it as part of the new divisor. In this case, 2 × 8 = 16.

Step 6: Find a digit n such that (160 + n) × n is less than or equal to 500. Here, n is 3 since 163 × 3 = 489.

Step 7: Subtract 489 from 500 to get a remainder of 11.

Step 8: Repeat the steps until the decimal reaches the desired precision.

The approximate value of √69 is 8.306.

The approximation method is another simple way to find the square roots of a number. Here's how to find the square root of 69 using the approximation method:

Step 1: Find the closest perfect squares around 69. The smaller perfect square is 64 and the larger is 81. √69 falls between √64 = 8 and √81 = 9.

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

Applying the formula: (69 - 64) / (81 - 64) = 5 / 17 ≈ 0.294.

Add this decimal to 8, the square root of 64.

Thus, 8 + 0.294 ≈ 8.294.

Therefore, √69 is approximately 8.294.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. For example, √9 = 3, since 3 × 3 = 9.

• Irrational number: An irrational number cannot be expressed as a simple fraction, such as π or √69.

• Radical form: A way of expressing roots, such as square or cube roots, using the radical symbol (√).

• Approximation: The process of finding a value that is close enough to the correct answer, typically used for non-perfect squares.

• Perfect square: A number that is the square of an integer, such as 64 (8 × 8) or 81 (9 × 9).