Square Root of 695

The square root is the inverse of the square of the number. 695 is not a perfect square. The square root of 695 is expressed in both radical and exponential form. In the radical form, it is expressed as √695, whereas (695)^(1/2) in the exponential form. √695 ≈ 26.366, 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 long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 695 is broken down into its prime factors:

Step 1: Finding the prime factors of 695

Breaking it down, we get 5 x 139, which means 695 = 5^1 x 139^1

Step 2: Now we found out the prime factors of 695. The second step is to make pairs of those prime factors. Since 695 is not a perfect square, the digits of the number can’t be grouped into pairs. Therefore, calculating √695 using prime factorization is not feasible.

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 with, we need to group the numbers from right to left. In the case of 695, we need to group it as 6 and 95.

Step 2: Now we need to find n whose square is ≤ 6. We can say n is ‘2’ because 2 x 2 = 4, which is less than 6. Now the quotient is 2 after subtracting 4 from 6, the remainder is 2.

Step 3: Bring down the next pair, 95, making the new dividend 295.

Step 4: Double the quotient obtained (2) to get the new divisor, 4.

Step 5: Find a digit, say ‘x,’ such that 4x * x ≤ 295. The correct x is 6, since 46 * 6 = 276.

Step 6: Subtract 276 from 295, and the remainder is 19.

Step 7: Since the dividend is smaller than the divisor, we need to add a decimal point, allowing us to bring down two zeroes, making the new dividend 1900.

Step 8: The new divisor now becomes 526 (adding 6 to 46 and appending a digit), and we find an x such that 526x * x ≤ 1900. The correct x is 3, since 526 * 3 = 1578.

Step 9: Continue these steps until you have the desired level of precision. The square root of 695 is approximately 26.366.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 695 using the approximation method.

Step 1: Identify the closest perfect squares. The closest perfect square less than 695 is 676 and the one greater is 729. √695 falls somewhere between 26 and 27.

Step 2: Use the formula: (Given number - smaller perfect square) ÷ (Greater perfect square - smaller perfect square). (695 - 676) ÷ (729 - 676) = 19 ÷ 53 ≈ 0.358

Step 3: Add this decimal to the smaller square root: 26 + 0.358 = 26.358 Thus, the square root of 695 is approximately 26.358.

• Square root: A square root is the inverse of a square. For example, 4^2 = 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, but the positive square root is more prominent due to its uses in the real world. This is known as the principal square root.

• Prime factorization: Breaking down a number into its prime factors. For example, the prime factorization of 695 is 5 x 139.

• Decimal: If a number has a whole number and a fraction in a single number, it is called a decimal. For example, 26.366 is a decimal.