Square Root of 668

The square root is the inverse of the square of the number. 668 is not a perfect square. The square root of 668 is expressed in both radical and exponential form. In the radical form, it is expressed as √668, whereas (668)^(1/2) in exponential form. √668 ≈ 25.835, 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 typically used for non-perfect square numbers where methods like the long-division method and approximation method are preferred. 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 668 is broken down into its prime factors.

Step 1: Finding the prime factors of 668 Breaking it down, we get 2 x 2 x 167.

Step 2: Now we have found the prime factors of 668. The second step is to make pairs of those prime factors. Since 668 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating the exact square root of 668 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 668, we group it as 68 and 6.

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

Step 3: Bring down 68 to make it the new dividend, making it 268. Double the quotient to get the new divisor, which is 4.

Step 4: Find a number X such that 4X x X is less than or equal to 268. We find X = 6 because 46 x 6 = 276, which is too high. Trying X = 5, we find 45 x 5 = 225, which is correct.

Step 5: Subtract 225 from 268 to get 43. Since 43 is less than the divisor, we need to add a decimal point and bring down two zeros.

Step 6: With the new dividend as 4300, repeat the process of finding the next digit in the quotient. Continue these steps until reaching the desired decimal precision.

The square root of 668 is approximately 25.83.

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 668 using the approximation method.

Step 1: Identify the closest perfect squares around 668. The smallest perfect square less than 668 is 625 (25^2), and the largest perfect square greater than 668 is 676 (26^2). Therefore, √668 falls between 25 and 26.

Step 2: Apply the formula: (Given number - smaller perfect square) ÷ (Larger perfect square - smaller perfect square). Using the formula: (668 - 625) ÷ (676 - 625) = 43 ÷ 51 ≈ 0.843. Adding this to the smaller root: 25 + 0.843 = 25.843, so the square root of 668 is approximately 25.843.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, which is √16 = 4.
   
• 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.
   
• Principal square root: A number has both positive and negative square roots; however, the positive square root is more commonly used in practical applications, hence called the principal square root.
   
• Prime factorization: Breaking down a number into its basic prime factors. Example: The prime factorization of 668 is 2 x 2 x 167.
   
• Decimal approximation: A close numerical value for an irrational number. Example: √668 ≈ 25.835.