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127 LearnersLast updated on December 15, 2025
Square Root of -12
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 concept extends into complex numbers when dealing with negative numbers. Here, we will discuss the square root of -12.
What is the Square Root of -12?
The square root is the inverse of the square of the number.
Since -12 is a negative number, its square root is not a real number. Instead, it is expressed in terms of imaginary numbers.
The square root of -12 can be expressed as √(-12) = √(12) × √(-1) = 2√3i, where 'i' is the imaginary unit, defined as √(-1).
Finding the Square Root of -12
To find the square root of -12, we need to use complex numbers.
The square root of -12 can be expressed in terms of its real and imaginary components.
Here, we will explore the method to derive the square root of -12:
1. Identify the real square root of the positive component, 12.
2. Multiply the result by the imaginary unit 'i' to account for the negative sign.
Square Root of -12 by Prime Factorization Method
The prime factorization method is used to simplify the positive part of the number before introducing the imaginary unit.
Let's explore how this works with -12:
Step 1: Factorize the positive component, 12, into its prime factors. Breaking it down, we get 2 × 2 × 3 = 2² × 3.
Step 2: Express the square root of 12. √12 = √(2² × 3) = 2√3.
Step 3: Introduce the imaginary unit for the negative sign. √(-12) = √12 × √(-1) = 2√3 × i = 2√3i.
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Square Root of -12 by Long Division Method
The long division method is typically used for non-perfect square real numbers, but when dealing with negative numbers, we need to use complex number operations.
Therefore, the long division method is not applicable directly for negative numbers like -12.
Instead, we use the concept of imaginary numbers as previously discussed.
Square Root of -12 by Approximation Method
While the approximation method is used for real numbers, for complex numbers such as the square root of -12, we rely on the direct calculation involving imaginary numbers.
The approximation method is not applicable here since the result involves 'i', the imaginary unit.
Common Mistakes and How to Avoid Them in the Square Root of -12
Students often make mistakes when dealing with square roots of negative numbers.
Let us look at a few common mistakes and how to avoid them.
Mistake 1
Forgetting about the imaginary unit 'i'
It is crucial to remember that the square root of a negative number involves the imaginary unit 'i'.
For example, forgetting to include 'i' in √(-12) results in an incorrect answer. The correct expression is 2√3i.
Mistake 2
Confusing real and imaginary components
Students might mistakenly assume the square root is entirely real or entirely imaginary.
It's important to understand that √(-12) = 2√3i has both a real component (2√3) and an imaginary unit (i).
Mistake 3
Applying real number methods directly
Methods like the long division method apply only to real numbers.
Attempting to use these directly on negative numbers without considering imaginary numbers leads to errors.
Mistake 4
Incorrect factorization of the positive part
Accurate factorization of the positive part is essential.
Missteps in factorizing 12 would lead to incorrect results for √(-12).
Ensure correct factorization as 2² × 3 before introducing 'i'.
Mistake 5
Incorrect simplification of complex numbers
Mistakes often occur in simplifying complex numbers involving 'i'.
Students should practice simplifying expressions like 2√3i accurately to avoid errors.
Square Root of -12 Examples
Problem 1
What is the square root of -12 expressed in terms of 'i'?
The square root of -12 is 2√3i.
Explanation
Since -12 is a negative number, its square root involves the imaginary unit 'i'.
Simplifying the positive part, we get √12 = 2√3, and including 'i', we have √(-12) = 2√3i.
Problem 2
If 2โ3i is the square root of -12, what is its square?
-12
Explanation
To find the square, multiply 2√3i by itself:
(2√3i)² = (2√3)² × i²
= 12 × (-1)
= -12.
Problem 3
Express the square root of -12 in terms of the real and imaginary parts.
Real part: 0, Imaginary part: 2√3
Explanation
The square root of -12 is completely imaginary, so the real part is 0 and the imaginary part is 2√3.
Problem 4
How does the square root of -12 relate to the square root of 12?
The square root of -12 is the square root of 12 multiplied by 'i'.
Explanation
The square root of -12 can be expressed as √12 × √(-1)
= √12 × i
= 2√3i.
Problem 5
Is the square root of -12 a real number?
No, it is a complex number.
Explanation
The square root of a negative number involves the imaginary unit 'i', which makes it a complex number, not a real number.
FAQ on Square Root of -12
1.What is โ(-12) in terms of 'i'?
2.Can the square root of -12 be a real number?
3.Why does โ(-12) involve 'i'?
4.What is the principal square root of -12?
5.Is the square of 2โ3i equal to -12?
Important Glossaries for the Square Root of -12
- Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For negative numbers, it involves imaginary numbers.
- Imaginary unit 'i': 'i' is defined as the square root of -1 and is used to express square roots of negative numbers.
- Complex number: A complex number comprises a real part and an imaginary part, typically expressed as a + bi.
- Prime factorization: Breaking down a number into its basic prime factors. For example, the prime factorization of 12 is 2² × 3.
- Principal square root: In the context of complex numbers, it refers to the positive imaginary component of a negative number's square root.
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.
