123 LearnersLast updated on May 26th, 2025
Square Root of 784
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
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.
Mistake 1
Forgetting about the negative square root
It is important to remember that a number has both positive and negative square roots. However, we commonly consider only the principal (positive) square root.
For example, √784 = 28, but there is also -28, which should not be forgotten.
Mistake 2
Not adding the square root symbol in proper places
Not using the square root symbol correctly is another mistake students make. It's important to simplify the numbers inside the square root first.
For Example: √(4 + 9) = √13 is incorrect; the correct method is √(13) without simplifying further incorrectly.
Mistake 3
Confusing the square root with the cube root
Students often confuse cube roots with square roots. It's important to distinguish between them.
For example, √64 = 8, while ∛64 = 4.
Mistake 4
Making errors in long division
Finding a square root using the long division method involves many steps. Students might skip steps accidentally, leading to incorrect results. It's important to practice and follow each step carefully.
Square Root of 784 Examples
Problem 1
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.
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.
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?
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.
Problem 3
Calculate √784 x 5.
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
Problem 4
What will be the square root of (784 + 16)?
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.
Problem 5
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.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√784 + 30) = 2 × (28 + 30) = 2 × 58 = 116 units.
FAQ on Square Root of 784
1.What is √784 in its simplest form?
2.Mention the factors of 784.
3.Calculate the square of 784.
4.Is 784 a prime number?
5.784 is divisible by?
6.How does learning Algebra help students in United States make better decisions in daily life?
7.How can cultural or local activities in United States support learning Algebra topics such as Square Root of 784?
8.How do technology and digital tools in United States support learning Algebra and Square Root of 784?
9.Does learning Algebra support future career opportunities for students in United States?
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.
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