Square Root of 7900

The square root is the inverse of the square of a number. 7900 is not a perfect square. The square root of 7900 is expressed in both radical and exponential form. In radical form, it is expressed as √7900, whereas (7900)^(1/2) in exponential form. √7900 ≈ 88.888, 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 7900 is broken down into its prime factors.

Step 1: Finding the prime factors of 7900 Breaking it down, we get 2 x 2 x 5 x 5 x 79: 2² x 5² x 79¹

Step 2: Now we have found the prime factors of 7900. The second step is to make pairs of those prime factors. Since 7900 is not a perfect square, the digits of the number can’t be grouped in complete pairs. Therefore, calculating √7900 using prime factorization is not straightforward.

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 7900, we need to group it as 79|00.

Step 2: Now we need to find n whose square is less than or equal to 79. We can say n is 8 because 8 x 8 = 64, which is less than 79. Now the quotient is 8 after subtracting 64 from 79; the remainder is 15.

Step 3: Now let us bring down 00, which makes the new dividend 1500. Add the old divisor with the same number: 8 + 8 = 16, which will be our new divisor.

Step 4: Now, find n such that 16n × n ≤ 1500. Let us consider n as 9; now 169 x 9 = 1521, which is larger than 1500, so we consider n as 8.

Step 5: Subtract 1500 from 1456 (168 x 8 = 1456); the difference is 44, and the quotient is 88.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 4400.

Step 7: Now we need to find the new divisor, which will be 176. Let’s find n such that 176n x n ≤ 4400. Suppose n is 2, then 1762 x 2 = 3524.

Step 8: Subtract 3524 from 4400, we get 876.

Step 9: Now the quotient is 88.8

Step 10: 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 √7900 ≈ 88.88

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 7900 using the approximation method.

Step 1: Now we have to find the closest perfect square of √7900. The smallest perfect square less than 7900 is 7744 (88^2) and the largest perfect square greater than 7900 is 7921 (89^2). √7900 falls somewhere between 88 and 89.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (7900 - 7744) ÷ (7921 - 7744) = 156 / 177 ≈ 0.881 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: 88 + 0.881 ≈ 88.881, so the square root of 7900 is approximately 88.881.

• Square root: A square root is the inverse of a square. For example, 5² = 25 and the inverse of the square is the square root, so √25 = 5.
   
• 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; the principal square root is the positive one, which is most often used in real-world applications.
   
• Prime factorization: Breaking down a number into the product of its prime factors. For example, the prime factorization of 90 is 2 × 3² × 5.
   
• Decimal: A decimal is a number that includes a whole number and a fractional part separated by a decimal point, such as 3.14 or 88.88.