Square Root of 7700

The square root is the inverse of the square of a number. 7700 is not a perfect square. The square root of 7700 is expressed in both radical and exponential form. In the radical form, it is expressed as √7700, whereas 7700^(1/2) is the exponential form. √7700 ≈ 87.7496, 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 7700 is broken down into its prime factors.

Step 1: Finding the prime factors of 7700 Breaking it down, we get 2 x 2 x 5 x 5 x 7 x 11: 2^2 x 5^2 x 7 x 11

Step 2: Now we found the prime factors of 7700. The second step is to make pairs of those prime factors. Since 7700 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 7700 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 7700, we need to group it as 77 and 00.

Step 2: Now we need to find n whose square is less than or equal to 77. We can say n is 8 because 8 x 8 = 64, which is less than 77. Now the quotient is 8, and the remainder is 77-64 = 13.

Step 3: Now let us bring down 00, making the new dividend 1300. Add the previous divisor and quotient, 8+8, to get 16, which is the new partial divisor.

Step 4: Now find the greatest digit x such that 16x x ≤ 1300. The correct digit is 7 because 167 x 7 = 1169, which is less than 1300.

Step 5: Subtract 1169 from 1300; the remainder is 131, and the quotient is 87.

Step 6: Since the dividend is less than the divisor, add a decimal point, allowing us to add two zeroes to the dividend. Now the new dividend is 13100.

Step 7: Continue the process to get two decimal places. The approximate square root of 7700 by long division is 87.75.

The approximation method is another method for finding square roots, and 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 7700 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √7700. The smallest perfect square less than 7700 is 7744, and the largest perfect square less than 7700 is 7569. √7700 falls somewhere between 87 and 88.

Step 2: Now we need to apply the formula that is (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). Using the formula: (7700 - 7569) / (7744 - 7569) = 131 / 175 ≈ 0.7486. Adding this to the smaller base: 87 + 0.7486 = 87.7486. So the square root of 7700 is approximately 87.75.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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. This is why it is also known as the principal square root.
   
• Decimals: If a number has a whole number and a fraction in a single number, then it is called a decimal, for example, 7.86, 8.65, and 9.42 are decimals
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• Long Division Method: A method used to find the square root of a number by dividing the number into groups of two digits, starting from the decimal point and working outwards.