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 the field of vehicle design, finance, etc. Here, we will discuss the square root of -24.
The square root is the inverse of the square of the number. Since -24 is a negative number, its square root is not a real number. The square root of -24 is expressed in terms of imaginary numbers. In the radical form, it is expressed as √(-24), which can be simplified to 2i√6, where "i" is the imaginary unit and equals √(-1).
Imaginary numbers are used when dealing with the square roots of negative numbers. The square root of -24 can be simplified by separately considering the square root of 24 and the imaginary unit 'i'. Let us explore the methods:
When dealing with the square root of negative numbers, we use the imaginary unit 'i'. Here is how we approach the square root of -24:
Step 1: Express -24 as a product: -24 = -1 × 24.
Step 2: The square root of -24 is √(-1 × 24) = √-1 × √24 = i × √24.
Step 3: Simplify √24 using prime factorization: 24 = 2 × 2 × 2 × 3 = 2² × 2 × 3.
Step 4: The square root of 24 is √(2² × 2 × 3) = 2√6.
Step 5: Therefore, the square root of -24 is 2i√6.
The prime factorization method helps in simplifying the square root of positive numbers like 24:
Step 1: Finding the prime factors of 24.
Breaking it down, we get 2 × 2 × 2 × 3: 2² × 2 × 3.
Step 2: Simplify the square root of 24 using its prime factors. The square root of 24 is √(2² × 2 × 3) = 2√6.
Step 3: Combine with the imaginary unit for -24.
So, the square root of -24 is 2i√6.
Understanding the properties of square roots involving negative numbers is crucial:
Students often make errors when dealing with the square roots of negative numbers due to the involvement of imaginary numbers. Here are some common mistakes:
Calculate the square root of -24.
2i√6
The square root of -24 is calculated by first recognizing it as √(-1 × 24), which becomes i√24.
Simplifying √24 gives 2√6.
Therefore, √(-24) = 2i√6.
What is the square of 2i√6?
-24
The square of 2i√6 is calculated as (2i√6)² = 4i² × 6 = 4 × -1 × 6 = -24 because i² = -1.
If x = √(-24), then what is x²?
-24
Given x = √(-24) = 2i√6, then x² = (2i√6)² = -24.
Find the product of 3 and the square root of -24.
6i√6
The square root of -24 is 2i√6.
Therefore, 3 × √(-24) = 3 × 2i√6 = 6i√6.
What is the square root of -24 squared?
24
The square root of -24 squared is |24| = 24 because (√(-24))² = -24 and taking the modulus gives 24.
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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