Square Root of 784

The square root is the inverse operation of squaring a number. 784 is a perfect square. The square root of 784 is expressed in both radical and exponential forms. In radical form, it is expressed as √784, whereas 784^(1/2) in exponential form. √784 = 28, which is a rational number because it can be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method can be used for perfect square numbers like 784. For non-perfect squares, methods such as 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 784 is broken down into its prime factors.

Step 1: Finding the prime factors of 784 Breaking it down, we get 2 x 2 x 2 x 2 x 7 x 7: 2^4 x 7^2

Step 2: Now that we have found the prime factors of 784, the next step is to make pairs of those prime factors. Since 784 is a perfect square, the digits of the number can be grouped in pairs. √784 = √(2^4 x 7^2) = 2^2 x 7 = 28

The long division method is particularly used for non-perfect square numbers, but it can also be used for perfect squares. Let's 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 pairs. In the case of 784, we pair them as 84 and 7.

Step 2: Find the largest number whose square is less than or equal to 7. We can say this number is 2 because 2 x 2 = 4 is less than 7. Now, the quotient is 2, and after subtracting 4 from 7, the remainder is 3.

Step 3: Bring down the next pair 84 to make it 384. Double the quotient (2), giving us 4, which will be our new divisor's initial part.

Step 4: Find a number n such that 4n x n ≤ 384. We find that n is 8, because 48 x 8 = 384. Subtract 384 from 384, and the remainder is 0.

Step 5: The quotient, therefore, is 28, indicating that √784 = 28.

The approximation method is useful for estimating the square roots of non-perfect squares. However, since 784 is a perfect square, we can straightforwardly find its square root.

Step 1: Find the perfect squares around √784. As it is a perfect square, we know √784 = 28 directly without further approximation.

Students often make mistakes while finding square roots, such as ignoring the negative square root or misapplying methods. Let us look at a few common mistakes in detail.

• Square root: A square root is the inverse operation of squaring a number. For example: 4^2 = 16 and the inverse of the square is the square root, that is √16 = 4.

• Rational number: A rational number is a number that can 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, the positive square root is often used, known as the principal square root.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 784 is a perfect square because it equals 28^2.

• Long division method: A method used to find square roots of numbers by performing division step-by-step, useful for both perfect and non-perfect squares.