Square Root of 7840

The square root is the inverse of the square of the number. 7840 is not a perfect square. The square root of 7840 is expressed in both radical and exponential form. In the radical form, it is expressed as √7840, whereas \(7840^{1/2}\) in the exponential form. √7840 ≈ 88.515, 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 7840 is broken down into its prime factors.

Step 1: Finding the prime factors of 7840 Breaking it down, we get 2 × 2 × 2 × 2 × 5 × 7 × 7 × 5: \(2^4 \times 5^2 \times 7^2\)

Step 2: Now we found out the prime factors of 7840. The second step is to make pairs of those prime factors. Since 7840 is not a perfect square, therefore the digits of the number can’t be grouped in pairs completely. Therefore, calculating 7840 using prime factorization can give us an 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 7840, we need to group it as 40 and 78.

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

Step 3: Now let us bring down 40 which is the new dividend. Add the old divisor with the same number: 8 + 8 to get 16, which will be our new divisor.

Step 4: The new divisor is now 16. We need to append a digit to make it suitable for dividing 1400.

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

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

Step 7: Since the new dividend is smaller, we need to add a decimal point, allowing us to add two zeroes to the dividend. Now the new dividend is 5600.

Step 8: Now we need to find the new divisor that is 177 because 177 × 3 = 531.

Step 9: Subtracting 531 from 5600 we get the result 69.

Step 10: Now the quotient is 88.5

Step 11: Continue these steps until the desired decimal precision is achieved. So the square root of √7840 is approximately 88.515.

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 7840 using the approximation method.

Step 1: Now we have to find the closest perfect square of √7840. The perfect squares surrounding 7840 are 7744 (which is 88²) and 7921 (which is 89²), so √7840 falls somewhere between 88 and 89.

Step 2: Now we need to apply the formula: \((\text{Given number} - \text{smallest perfect square}) / (\text{Greater perfect square} - \text{smallest perfect square})\). Using the formula \((7840 - 7744) / (7921 - 7744) = 96/177 = 0.542\). Using the formula, we identified the decimal part of our square root. The next step is adding the initial integer we got to the decimal number: 88 + 0.542 = 88.542, so the square root of 7840 is approximately 88.542.

• 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, that is, \(\sqrt{16} = 4\).
   
• 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 iteratively calculating digits of the root.
   
• Approximation method: A method for estimating the square root of a number by identifying the nearest perfect squares and interpolating between them.