Square Root of 634

The square root is the inverse of squaring a number. 634 is not a perfect square. The square root of 634 is expressed in both radical and exponential form. In the radical form, it is expressed as √634, whereas in exponential form it is (634)^(1/2). √634 ≈ 25.192, 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, the 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 634 is broken down into its prime factors:

Step 1: Finding the prime factors of 634 Breaking it down, we get 2 x 317: 2^1 x 317^1

Step 2: Now we found out the prime factors of 634. The next step is to make pairs of those prime factors. Since 634 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 634 using prime factorization results in an approximation.

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 634, we group it as 34 and 6.

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

Step 3: Bring down 34 to make the new dividend 234.

Step 4: Add the previous divisor with the same number, 2 + 2 = 4, which will be our new divisor.

Step 5: Find 4n × n ≤ 234. Let us consider n as 5, now 45 x 5 = 225.

Step 6: Subtract 234 from 225; the remainder is 9, and the quotient is 25.

Step 7: Since the dividend is less than the divisor, add a decimal point and bring down two zeros to make it 900.

Step 8: Find the new divisor; 50 because 501 x 1 = 501.

Step 9: Subtract 501 from 900 to get a remainder of 399.

Step 10: The quotient is 25.1

Step 11: Continue these steps until you achieve the desired decimal precision.

So the square root of √634 is approximately 25.192

The approximation method is another method for finding the 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 634 using the approximation method.

Step 1: Find the closest perfect squares around √634 The closest perfect squares are 625 and 676, with √634 falling between 25 and 26.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (634 - 625) / (676 - 625) = 9 / 51 ≈ 0.176 Add the approximation to the smaller square root: 25 + 0.176 = 25.176, so the square root of 634 is approximately 25.176.

• Square root: A square root is the inverse of squaring a number. For example, if 5^2 = 25, then √25 = 5.
   
• Irrational number: An irrational number cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Perfect square: A number that is the square of an integer. For example, 36 is a perfect square because it is 6^2.
   
• Decimal: A number expressed in the scale of tens, often used to represent non-whole numbers. For example, 25.192 is a decimal.
   
• Approximation: An approximate calculation or estimation of a value, often used in calculating non-perfect squares.