Square Root of 7956

The square root is the inverse of the square of a number. 7956 is not a perfect square. The square root of 7956 is expressed in both radical and exponential form. In the radical form, it is expressed as √7956, whereas (7956)^(1/2) in the exponential form. √7956 ≈ 89.1922, 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 7956 is broken down into its prime factors.

Step 1: Finding the prime factors of 7956 Breaking it down, we get 2 × 2 × 3 × 3 × 3 × 37: 2² × 3³ × 37

Step 2: Now we found out the prime factors of 7956. The second step is to make pairs of those prime factors. Since 7956 is not a perfect square, the digits of the number can’t be grouped perfectly into pairs. Therefore, calculating 7956 using prime factorization is challenging without further 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 7956, we need to group it as 56 and 79.

Step 2: Now we need to find n whose square is close to or less than 79. We can say n is ‘8’ because 8 × 8 = 64, which is less than or equal to 79. Now the quotient is 8, and after subtracting 79 - 64, the remainder is 15.

Step 3: Now let us bring down 56, which is the new dividend. Add the old divisor with the same number, 8 + 8 = 16, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 16n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 16n × n ≤ 1556. Let us consider n as 9, now 16 × 9 = 144.

Step 6: Subtract 1556 from 144, and the difference is 1256, and the quotient is 89.

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 125600.

Step 8: Now we need to find a new divisor that is 892 because 892 × 2 = 1784, which is less than 12560.

Step 9: Subtracting 1784 from 12560, we get the result 10776.

Step 10: Now the quotient is 89.1

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero. So the square root of √7956 is approximately 89.1922.

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 7956 using the approximation method.

Step 1: Now we have to find the closest perfect square of √7956. The smallest perfect square less than 7956 is 7921, and the nearest perfect square greater than 7956 is 8100. √7956 falls somewhere between 89 and 90.

Step 2: Now we need to apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (7956 - 7921) / (8100 - 7921) = 35 / 179 ≈ 0.1955

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 89 + 0.1955 ≈ 89.1955, so the square root of 7956 is approximately 89.1955.

• Square root: A square root is the inverse of a square. For example, 4² = 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; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors. For example, the prime factorization of 18 is 2 × 3².
   
• Long division method: The long division method is a technique used to find the square root of a non-perfect square by dividing and approximating the root step by step.