Square Root of 7800

The square root is the inverse of the square of the number. 7800 is not a perfect square. The square root of 7800 is expressed in both radical and exponential form. In the radical form, it is expressed as √7800, whereas (7800)^(1/2) in the exponential form. √7800 ≈ 88.313, 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 7800 is broken down into its prime factors.

Step 1: Finding the prime factors of 7800 Breaking it down, we get 2 x 2 x 2 x 3 x 5 x 5 x 13: 2^3 x 3^1 x 5^2 x 13^1

Step 2: Now we found the prime factors of 7800. The second step is to make pairs of those prime factors. Since 7800 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √7800 using prime factorization is impossible.

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 7800, we need to group it as 80 and 78.

Step 2: Now we need to find n whose square is close to 78. We can say n is ‘8’ because 8 x 8 = 64, which is less than 78. Now the quotient is 8, after subtracting 78 - 64, the remainder is 14.

Step 3: Now bring down 00, which is the new dividend. Add the old divisor with the same number, 8 + 8, we get 16, which will be our new divisor.

Step 4: The new divisor will be 16n. We need to find the value of n.

Step 5: The next step is finding 16n x n ≤ 1400. Let us consider n as 8, now 168 x 8 = 1344.

Step 6: Subtract 1344 from 1400. The difference is 56, and the quotient is 88.

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 5600.

Step 8: Now we need to find the new divisor, which is 176, because 1763 x 3 = 5289.

Step 9: Subtracting 5289 from 5600, we get the result 311.

Step 10: Now the quotient is 88.3.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero. So the square root of √7800 is 88.31.

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 7800 using the approximation method.

Step 1: Now we have to find the closest perfect square of √7800. The smallest perfect square less than 7800 is 7744 (which is 88^2), and the largest perfect square greater than 7800 is 7921 (which is 89^2). √7800 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 (7800 - 7744) / (7921 - 7744) = 56 / 177 ≈ 0.316 Adding this decimal to 88 gives us 88 + 0.316 = 88.316, so the square root of 7800 is approximately 88.316.

• Square root: A square root is the inverse of a square. For example, 4^2 = 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 the principal square root.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors.
   
• Approximation: Approximation is a method of estimating a value based on nearby known values. It is often used to find the square root of non-perfect squares.