Square Root of 672

The square root is the inverse of the square of the number. 672 is not a perfect square. The square root of 672 is expressed in both radical and exponential form. In the radical form, it is expressed as √672, whereas (672)^(1/2) in the exponential form. √672 ≈ 25.922, 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 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 672 is broken down into its prime factors:

Step 1: Finding the prime factors of 672 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 7: 2^4 x 3 x 7

Step 2: Now we found out the prime factors of 672. The second step is to make pairs of those prime factors. Since 672 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 672 using prime factorization is impossible for finding an exact square root.

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 672, we need to group it as 72 and 6.

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

Step 3: Now let us bring down 72, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be followed by finding the value of n such that 4n x n ≤ 272.

Step 5: The next step is finding 4n x n ≤ 272. Let us consider n as 6; now 46 x 6 = 276, which is too large, so we try n as 5.

Step 6: Subtract 245 (45 x 5) from 272; the difference is 27, and the quotient is 25.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 2700.

Step 8: We need to find the new divisor, which is 509 because 509 x 5 = 2545.

Step 9: Subtracting 2545 from 2700, we get the result 155.

Step 10: Now the quotient is 25.9.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values; continue until the remainder is zero.

So the square root of √672 is approximately 25.92.

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 672 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √672. The smallest perfect square less than 672 is 625, and the largest perfect square greater than 672 is 729. √672 falls somewhere between 25 and 26.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula: (672 - 625) ÷ (729 - 625) = 47 ÷ 104 ≈ 0.452. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 25 + 0.452 = 25.452, so the square root of 672 is approximately 25.92.

• Square root: A square root is the inverse of squaring a number. Example: 4^2 = 16, and the inverse of the square is the square root, i.e., √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 usually taken as it is more commonly used in real-world applications. This is known as the principal square root.
   
• Prime factorization: Prime factorization is expressing a number as a product of its prime factors.
   
• Perfect square: A perfect square is a number that can be expressed as the product of an integer with itself. For example, 25 is a perfect square because it is 5 x 5.