BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon101 Learners

Last updated on May 26th, 2025

Math Whiteboard Illustration

Square Root of -784

Professor Greenline Explaining Math Concepts

If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in fields like engineering, physics, and complex analysis. Here, we will discuss the square root of -784.

Square Root of -784 for Saudi Students
Professor Greenline from BrightChamps

What is the Square Root of -784?

The square root is the inverse of the square of a number. Since -784 is negative, its square root is not a real number. Instead, it is an imaginary number. The square root of -784 is expressed in terms of the imaginary unit, i, which is defined as √-1. Thus, the square root of -784 is written as √-784 = √784 * √-1 = 28i. Because it involves i, it is an imaginary number and not a real or rational number.

Professor Greenline from BrightChamps

Finding the Square Root of -784

The prime factorization method is not applicable for negative numbers. Instead, we consider the absolute value of the number and use imaginary numbers. Here are the steps:

 

  • Take the absolute value of -784, which is 784.
  • Find the square root of 784, which is a perfect square: √784 = 28.
  • Include the imaginary unit i to account for the negative sign: √-784 = 28i.
Professor Greenline from BrightChamps

Square Root of -784 by Prime Factorization Method

Since -784 is negative, we consider the prime factorization of its absolute value, 784. Here's how it's done:

 

Step 1: Find the prime factors of 784.

 

Breaking it down, we have 2 × 2 × 2 × 2 × 7 × 7, which simplifies to 2⁴ × 7².

 

Step 2: Pair the prime factors. The pairings are (2²)² and (7¹)², making 784 a perfect square.

 

Step 3: Combine the pairs to find the square root, which is 28.

 

Step 4: Attach the imaginary unit i to account for the negative sign, resulting in √-784 = 28i.

Professor Greenline from BrightChamps

Square Root of -784 by Long Division Method

The long division method is typically used for non-perfect square numbers. Here, we use it to confirm the real part, then apply the imaginary unit.

 

Step 1: Start with the absolute value, 784, and find its square root, using long division if needed.

 

Step 2: For 784, the perfect square is 28.

 

Step 3: Finally, apply the imaginary unit: √-784 = 28i.

Professor Greenline from BrightChamps

Square Root of -784 by Approximation Method

This method is useful if precise calculation of a non-perfect square is needed, but 784 is a perfect square. Thus, we can directly calculate:

 

Step 1: Determine the nearest perfect squares around 784, but since 784 is a perfect square, √784 = 28.

 

Step 2: Include the imaginary unit to account for the negative sign: √-784 = 28i.

Max Pointing Out Common Math Mistakes

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

Students often make mistakes with negative numbers and imaginary units. Understanding these concepts is crucial. Let's explore some common mistakes and how to avoid them.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Ignoring the Imaginary Unit

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

One common mistake is forgetting the imaginary unit when dealing with negative square roots. Always remember that the square root of a negative number involves i, the imaginary unit.

 

For example, √-4 is 2i, not just 2.

Max from BrightChamps Saying "Hey"

Square Root of -784 Examples

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Can you help Max find the area of a square box if its side length is given as √-196?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The area of the square is -196 square units, noting it involves imaginary numbers.

Explanation

The area of the square = side².

The side length is given as √-196 = 14i.

Area of the square = (14i)² = 196i² = -196 (since i² = -1).

Therefore, the area of the square box is -196 square units.

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 2

A square-shaped plot measuring -784 square feet is envisaged; if each of the sides is √-784, what will be the square feet of half of the plot?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

-392 square feet

Explanation

Divide the imaginary area by 2, but remember the imaginary nature.

Dividing -784 by 2 gives -392.

So, half of the plot measures -392 square feet, considering the imaginary aspect.

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 3

Calculate √-784 x 5.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

140i

Explanation

The first step is to find the square root of -784, which is 28i, then multiply by 5.

So, 28i x 5 = 140i.

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 4

What will be the square root of (-400 + 16)?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The square root is ±20i

Explanation

To find the square root, calculate (-400 + 16) = -384.

Then, √-384 = √384 * i = 20i (assuming simplification though √384 isn't perfect).

Therefore, the square root of (-400 + 16) is ±20i.

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 5

Find the perimeter of the square if its side ‘s’ is √-784 units.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The perimeter is 112i units.

Explanation

Perimeter of a square = 4 × side

Perimeter = 4 × √-784 = 4 × 28i = 112i units.

Max from BrightChamps Praising Clear Math Explanations
Ray Thinking Deeply About Math Problems

FAQ on Square Root of -784

1.What is √-784 in its simplest form?

Math FAQ Answers Dropdown Arrow

2.Is -784 a perfect square?

Math FAQ Answers Dropdown Arrow

3.Calculate the square of 784.

Math FAQ Answers Dropdown Arrow

4.What is the imaginary unit?

Math FAQ Answers Dropdown Arrow

5.How do you express -784 as a product of prime factors?

Math FAQ Answers Dropdown Arrow

6.How does learning Algebra help students in Saudi Arabia make better decisions in daily life?

Math FAQ Answers Dropdown Arrow

7.How can cultural or local activities in Saudi Arabia support learning Algebra topics such as Square Root of -784?

Math FAQ Answers Dropdown Arrow

8.How do technology and digital tools in Saudi Arabia support learning Algebra and Square Root of -784?

Math FAQ Answers Dropdown Arrow

9.Does learning Algebra support future career opportunities for students in Saudi Arabia?

Math FAQ Answers Dropdown Arrow
Professor Greenline from BrightChamps

Important Glossaries for the Square Root of -784

  • Imaginary Number: A number that involves the imaginary unit i, representing the square root of negative numbers. Example: √-9 = 3i.

 

  • Complex Number: A combination of real and imaginary numbers in the form a + bi, where a and b are real numbers.

 

  • Perfect Square: A number that is the square of an integer. Example: 784 is a perfect square because 28² = 784.

 

  • Imaginary Unit (i): Defined as √-1, it is used to express complex numbers involving square roots of negative numbers.

 

  • Square Root: The inverse operation of squaring a number, involving both real and imaginary roots for negative numbers.
Professor Greenline from BrightChamps

About BrightChamps in Saudi Arabia

At BrightChamps, we recognize algebra as more than just symbols—it’s a key to unlock countless opportunities! Our goal is to help children across Saudi Arabia gain important math skills, focusing today on the Square Root of -784 with special attention to square roots—in a way that’s engaging, lively, and easy to grasp. Whether your child is calculating the speed of a roller coaster at Riyadh’s Al Hokair Land, following scores at local football matches, or managing their allowance for the latest gadgets, mastering algebra boosts their confidence for daily challenges. Our interactive lessons make learning accessible and fun. Since children in Saudi Arabia learn in different ways, we tailor lessons to suit each learner. From Riyadh’s bustling streets to Jeddah’s historic landmarks, BrightChamps brings math to life, making it exciting and relevant all over Saudi Arabia. Let’s make square roots a fun part of every child’s math adventure!
Math Teacher Background Image
Math Teacher Image

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.

Math Teacher Fun Facts Image
Max, the Girl Character from BrightChamps

Fun Fact

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

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
Dubai - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom