104 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.
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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.
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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.
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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?
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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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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.
