Square Root of 7825

The square root is the inverse of the square of a number. 7825 is not a perfect square. The square root of 7825 is expressed in both radical and exponential form. In the radical form, it is expressed as √7825, whereas (7825)^(1/2) in the exponential form. √7825 ≈ 88.436, 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 7825 is broken down into its prime factors.

Step 1: Finding the prime factors of 7825 Breaking it down, we get 5 x 5 x 313: \(5^2 \times 313^1\)

Step 2: Now we found out the prime factors of 7825. The second step is to make pairs of those prime factors. Since 7825 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 7825 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 7825, we group it as 78 and 25.

Step 2: Now we need to find n whose square is 64. We can say n as ‘8’ because 8 x 8 is lesser than or equal to 78. Now the quotient is 8 after subtracting 64 from 78, the remainder is 14

Step 3: Now let us bring down 25 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 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 x n ≤ 1425. Let us consider n as 8, now 16 x 8 + 8 = 136 Step 6: Subtract 136 from 1425, the difference is 89, 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 zeros to the dividend. Now the new dividend is 8900.

Step 8: Now we need to find the new divisor that is 883 because 883 x 10 = 8830.

Step 9: Subtracting 8830 from 8900 we get the result 70.

Step 10: Now the quotient is 88.4.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values, continue till the remainder is zero. So the square root of √7825 ≈ 88.44.

The approximation method is another method for finding the 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 7825 using the approximation method.

Step 1: Now we have to find the closest perfect square of √7825. The smallest perfect square near 7825 is 7744 and the largest perfect square near 7825 is 7921. √7825 falls somewhere between 88 and 89.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (7825 - 7744) / (7921 - 7744) = 81 / 177 = 0.457 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 88 + 0.457 = 88.457, so the square root of 7825 is approximately 88.457.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, which 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.
   
• Long division method: A method used to find the square root of non-perfect squares by a series of division steps to approximate the root value.
   
• Approximation method: A technique to estimate the square root of a number by using nearby perfect squares to find a closer value.