Last updated on May 26th, 2025
If a number is multiplied by itself, the result is a square. The inverse of squaring is finding the square root. The square root is used in various fields such as architecture, engineering, and finance. Here, we will discuss the 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:
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.
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in methods. Now let us look at a few of those mistakes that students tend to make in detail.
Can you help Max find the area of a square box if its side length is given as √784?
The area of the square is 784 square units.
The area of the square = side^2.
The side length is given as √784.
Area of the square = side^2 = √784 x √784 = 28 x 28 = 784
Therefore, the area of the square box is 784 square units.
A square-shaped building measuring 784 square feet is built; if each of the sides is √784, what will be the square feet of half of the building?
392 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 784 by 2 = we get 392
So half of the building measures 392 square feet.
Calculate √784 x 5.
140
The first step is to find the square root of 784, which is 28.
Multiply 28 by 5. So 28 x 5 = 140
What will be the square root of (784 + 16)?
The square root is 28.57 (approximately)
To find the square root, we need to find the sum of (784 + 16). 784 + 16 = 800, and then √800 ≈ 28.57.
Therefore, the square root of (784 + 16) is approximately ±28.57.
Find the perimeter of the rectangle if its length 'l' is √784 units and the width 'w' is 30 units.
We find the perimeter of the rectangle to be 116 units.
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√784 + 30) = 2 × (28 + 30) = 2 × 58 = 116 units.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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