112 LearnersLast updated on May 26th, 2025
Square Root of -104
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The concept of square roots extends to complex numbers, where we can find the square root of negative numbers. Here, we will discuss the square root of -104.
What is the Square Root of -104?
The square root is the inverse of the square of a number. Since -104 is a negative number, its square root is a complex number. The square root of -104 can be expressed in terms of the imaginary unit i, where i² = -1. Therefore, the square root of -104 is expressed as √-104 = √(104) × i = 10.198 × i, which is a complex number.
Finding the Square Root of -104
To find the square root of a negative number, the concept of complex numbers is used. The steps involve finding the square root of the absolute value and multiplying by the imaginary unit i.
Step 1: Find the square root of the absolute value of -104, which is 104.
Step 2: √104 = 10.198.
Step 3: Multiply the result by i to get the square root of -104: √-104 = 10.198 × i.
Square Root of -104 by Prime Factorization Method
The prime factorization method is not applicable directly for negative numbers, but we can find the prime factorization of 104 first.
Step 1: Finding the prime factors of 104: 104 = 2 × 2 × 2 × 13.
Step 2: The prime factors of 104 are 2³ × 13.
Step 3: The square root of 104 in radical form is simplified as √(2³ × 13).
Step 4: The square root of -104 is then √(2³ × 13) × i.
Square Root of -104 by Long Division Method
The long division method is used for finding the square root of non-perfect square positive numbers. For negative numbers, we first find the square root of the absolute value and then multiply by i.
Step 1: Apply the long division method to find √104.
Step 2: Group the digits of 104 as (1)(04).
Step 3: Find the largest number n whose square is ≤1. It is 1.
Step 4: Subtract 1² from 1 and bring down 04, making it 104.
Step 5: Double the divisor and find the next digit.
Step 6: Continue the process to find √104 ≈ 10.198. Step 7: Multiply by i to get √-104 ≈ 10.198 × i.
Square Root of -104 by Approximation Method
The approximation method involves estimating the square root by finding nearby perfect squares.
Step 1: Identify the closest perfect squares to 104, which are 100 and 121.
Step 2: √100 = 10, and √121 = 11, so √104 is between 10 and 11.
Step 3: Estimate √104 ≈ 10.198.
Step 4: Multiply by i for the square root of -104: √-104 ≈ 10.198 × i.
Common Mistakes and How to Avoid Them in Finding the Square Root of -104
Mistakes can occur when finding square roots, especially with negative numbers. Let’s look at some common mistakes students might make and how to avoid them.
Mistake 1
Ignoring the Imaginary Unit
Students often forget to include the imaginary unit i when dealing with negative square roots. Remember that the square root of a negative number involves i.
For instance, √-104 should be written as 10.198 × i, not just 10.198.
Square Root of -104 Examples
Problem 1
Can you help Max find the magnitude of a vector with a component of √-104?
The magnitude of the vector is 10.198.
Explanation
The magnitude of a vector is the absolute value of its components. Since √-104 = 10.198 × i, the magnitude is |10.198 × i| = 10.198.
Problem 2
A complex number is given as 3 + √-104. What is the modulus of this complex number?
The modulus is 10.63.
Explanation
The modulus of a complex number a + bi is √(a² + b²). Here, a = 3 and b = 10.198. Modulus = √(3² + 10.198²) = √(9 + 103.9924) = √112.9924 ≈ 10.63.
Problem 3
Calculate 2 × √-104.
The result is 20.396i.
Explanation
First, find the square root of -104: √-104 = 10.198 × i. Then multiply by 2: 2 × 10.198 × i = 20.396i.
Problem 4
What will be the square root of (-100 + -4)?
The square root is 10i.
Explanation
First, find the sum of (-100 + -4) = -104. Then, find √-104 = 10.198 × i, approximately equal to 10i for simplicity.
Problem 5
Find the expression for the square of √-104.
The expression is -104.
Explanation
Since the square root and squaring are inverse operations, (√-104)² = -104.
FAQ on Square Root of -104
1.What is √-104 in its simplest form?
2.Can we find real square roots of negative numbers?
3.What does the imaginary unit i represent?
4.Is -104 a perfect square?
5.What are complex numbers?
Important Glossaries for the Square Root of -104
- Complex Number: A number comprising a real part and an imaginary part, often written as a + bi.
- Imaginary Unit: Denoted as i, it represents √-1 and is used to express complex numbers.
- Absolute Value: The non-negative value of a number without regard to its sign, applicable to both real and complex numbers.
- Modulus: The magnitude or absolute value of a complex number, calculated as the square root of the sum of the squares of its real and imaginary parts.
- Imaginary Number: A number that can be written in the form of a real number multiplied by the imaginary unit i.
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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.
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