Square Root of -784
2026-02-28 13:37 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-784

201 Learners

Last updated on August 5, 2025

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.

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.

Finding the Square Root of -784

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.

Explore Our Programs

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.

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.

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.

Problem 1

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

Okay, lets 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.

Well explained 👍

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?

Okay, lets 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.

Well explained 👍

Problem 3

Calculate √-784 x 5.

Okay, lets 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.

Well explained 👍

Problem 4

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

Okay, lets 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.

Well explained 👍

Problem 5

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

Okay, lets begin

The perimeter is 112i units.

Explanation

Perimeter of a square = 4 × side

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

Well explained 👍

FAQ on Square Root of -784

1.What is √-784 in its simplest form?

The simplest form of √-784 is 28i, using the prime factorization 2⁴ × 7² for 784 and the imaginary unit i.

2.Is -784 a perfect square?

Yes, 784 is a perfect square, but -784 is not a real perfect square since it involves imaginary numbers.

3.Calculate the square of 784.

The square of 784 is 784 x 784 = 614656.

4.What is the imaginary unit?

The imaginary unit, denoted as i, is defined as the square root of -1. It is used in complex numbers.

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

The prime factorization of 784 is 2⁴ × 7². We don't factor -1 but use it to introduce the imaginary unit i.

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.

What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math

▶

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.