Square Root of 784
2026-02-28 13:28 Diff
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Last updated on August 5, 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.

What is 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.

Finding the Square Root of 784

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

Square Root of 784 by Prime Factorization 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

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Square Root of 784 by Long Division Method

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.

Square Root of 784 by Approximation Method

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.

Common Mistakes and How to Avoid Them in the Square Root of 784

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.

Common Mistakes and How to Avoid Them in the Square Root of 784

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.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √784?

Okay, lets begin

The area of the square is 784 square units.

Explanation

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.

Well explained 👍

Problem 2

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?

Okay, lets begin

392 square feet

Explanation

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.

Well explained 👍

Problem 3

Calculate √784 x 5.

Okay, lets begin

140

Explanation

The first step is to find the square root of 784, which is 28.

Multiply 28 by 5. So 28 x 5 = 140

Well explained 👍

Problem 4

What will be the square root of (784 + 16)?

Okay, lets begin

The square root is 28.57 (approximately)

Explanation

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.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length 'l' is √784 units and the width 'w' is 30 units.

Okay, lets begin

We find the perimeter of the rectangle to be 116 units.

Explanation

Perimeter of the rectangle = 2 × (length + width)

Perimeter = 2 × (√784 + 30) = 2 × (28 + 30) = 2 × 58 = 116 units.

Well explained 👍

FAQ on Square Root of 784

1.What is √784 in its simplest form?

The prime factorization of 784 is 2 x 2 x 2 x 2 x 7 x 7, so the simplest form of √784 = √(2^4 x 7^2) = 2^2 x 7 = 28.

2.Mention the factors of 784.

Factors of 784 are 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, and 784.

3.Calculate the square of 784.

We get the square of 784 by multiplying the number by itself, that is 784 x 784 = 614656.

4.Is 784 a prime number?

5.784 is divisible by?

784 has many factors; those are 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, and 784.

Important Glossaries for the Square Root of 784

  • 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.

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Jaskaran Singh Saluja

About the Author

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.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.