Last updated on May 26th, 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.
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.
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:
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.
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.
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.
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.
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.
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.
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
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.
Calculate √-784 x 5.
140i
The first step is to find the square root of -784, which is 28i, then multiply by 5.
So, 28i x 5 = 140i.
What will be the square root of (-400 + 16)?
The square root is ±20i
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.
Find the perimeter of the square if its side ‘s’ is √-784 units.
The perimeter is 112i units.
Perimeter of a square = 4 × side
Perimeter = 4 × √-784 = 4 × 28i = 112i units.
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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