111 LearnersLast updated on May 26th, 2025
Square Root of -784
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
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.
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.
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.
Mistake 1
Ignoring the Imaginary Unit
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.
Mistake 2
Misapplying the Square Root Properties
Another mistake is incorrectly applying square root properties, such as assuming √(a*b) = √a * √b without considering the sign. Always separate the negative sign and apply the imaginary unit correctly.
Mistake 3
Incorrect Factorization
Students may incorrectly factorize the number, leading to errors in finding its square root. Ensure factorization is done correctly, especially for perfect squares.
Mistake 4
Confusing Real and Imaginary Numbers
It's crucial to differentiate between real and imaginary numbers. Real numbers have no imaginary unit, while imaginary numbers do, such as 28i for √-784.
Mistake 5
Errors in Long Division
Long division can be complex, leading to mistakes. Ensure each step is followed carefully, especially when confirming perfect squares like 784, even if using imaginary numbers.
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 √-196?
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.
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?
-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.
Problem 3
Calculate √-784 x 5.
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.
Problem 4
What will be the square root of (-400 + 16)?
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.
Problem 5
Find the perimeter of the square if its side ‘s’ is √-784 units.
The perimeter is 112i units.
Explanation
Perimeter of a square = 4 × side
Perimeter = 4 × √-784 = 4 × 28i = 112i units.
FAQ on Square Root of -784
1.What is √-784 in its simplest form?
2.Is -784 a perfect square?
3.Calculate the square of 784.
4.What is the imaginary unit?
5.How do you express -784 as a product of prime factors?
6.How does learning Algebra help students in Saudi Arabia make better decisions in daily life?
7.How can cultural or local activities in Saudi Arabia support learning Algebra topics such as Square Root of -784?
8.How do technology and digital tools in Saudi Arabia support learning Algebra and Square Root of -784?
9.Does learning Algebra support future career opportunities for students in Saudi Arabia?
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.
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About BrightChamps in Saudi Arabia
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.
