113 LearnersLast updated on May 26th, 2025
Square Root of -48
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 into complex numbers for negative values. Here, we will discuss the square root of -48.
What is the Square Root of -48?
The square root of a negative number involves imaginary numbers. In this case, the square root of -48 is expressed in the form of an imaginary number. The square root of -48 can be represented as √(-48) = √(48) * √(-1) = 4√3 * i, where i is the imaginary unit, defined as √(-1).
Finding the Square Root of -48
The square root of a negative number is not a real number, so we use the concept of imaginary numbers. The process involves finding the square root of the positive part and multiplying it by the imaginary unit i.
Step 1: Identify the positive part of -48, which is 48.
Step 2: Find the square root of 48. The prime factorization of 48 is 2 x 2 x 2 x 2 x 3, which can be grouped as (2 x 2) x (2 x 2) x 3 = 4 x 4 x 3.
Step 3: The square root of 48 is 4√3, as 4 is the square root of 16 (which is 4 x 4), and √3 remains under the radical.
Step 4: Include the imaginary unit i, thus √(-48) = 4√3 * i.
Square Root of -48 by Prime Factorization Method
To find the square root of -48 using prime factorization, we focus on the positive part 48 and then include the imaginary unit:
Step 1: Find the prime factors of 48, which are 2 x 2 x 2 x 2 x 3.
Step 2: Pair the prime factors to simplify: (2 x 2) x (2 x 2) x 3 = 4 x 4 x 3.
Step 3: The square root of 48 becomes 4√3 since 4 is the square root of 16.
Step 4: Since we are dealing with -48, we multiply by i, hence √(-48) = 4√3 * i.
Square Root of -48 by Long Division Method
The long division method is typically used for non-perfect square numbers. However, when dealing with negative numbers, the focus is on the positive component:
Step 1: Start by considering the positive part, 48. Group as 48.
Step 2: Find the closest perfect square less than 48, which is 36. The square root of 36 is 6.
Step 3: Approximate further to find that √48 is approximately 6.9282.
Step 4: Combine with the imaginary unit i, so √(-48) = 6.9282 * i, which simplifies to 4√3 * i.
Square Root of -48 by Approximation Method
The approximation method involves estimating the square root using nearby perfect squares:
Step 1: The closest perfect squares around 48 are 36 (6 squared) and 49 (7 squared).
Step 2: √48 falls between 6 and 7. Calculate the approximation as follows:
Step 3: Using the formula (Given number - smaller perfect square) ÷ (larger perfect square - smaller perfect square), we find (48 - 36) ÷ (49 - 36) = 12 ÷ 13 ≈ 0.923.
Step 4: Add this to the smaller square root, 6 + 0.923 ≈ 6.923, and multiply by i, so √(-48) ≈ 6.923 * i = 4√3 * i.
Common Mistakes and How to Avoid Them in the Square Root of -48
Students often make errors when dealing with negative square roots, such as forgetting the role of the imaginary unit, i. Let's explore common mistakes and how to avoid them:
Mistake 1
Forgetting the Imaginary Unit
When finding the square root of a negative number, it is crucial to remember that the result is an imaginary number.
For example, √(-48) should be expressed as 4√3 * i, not simply as a real number.
Mistake 2
Incorrect Factorization
Errors in factorization can lead to incorrect simplification. Ensure correct grouping of factors for simplification, such as √(48) = 4√3, not 2√12.
Mistake 3
Confusing Real and Imaginary Parts
Students might confuse real numbers and imaginary numbers. Emphasize that the square root of negative numbers always involves the imaginary unit i.
Mistake 4
Ignoring the Complexity of Negative Roots
Some might attempt to simplify negative square roots as if they were positive. Ensure students understand that negative square roots require the use of the imaginary unit.
Mistake 5
Mixing Square Root and Cube Root Notations
It's essential to distinguish between square root (√) and cube root (∛) notations. Confusing these can lead to incorrect calculations and interpretations.
Square Root of -48 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √(-48)?
The area of the square is -48 square units.
Explanation
The area of the square = side^2.
The side length is given as √(-48) = 4√3 * i.
Area of the square = (4√3 * i)^2 = -48.
Therefore, the area of the square box is -48 square units.
Problem 2
A square-shaped building measuring -48 square feet is built; if each of the sides is √(-48), what will be the square feet of half of the building?
-24 square feet
Explanation
The area of the building is -48 square feet.
Half of this area is -24 square feet.
So half of the building measures -24 square feet.
Problem 3
Calculate 2 * √(-48).
8√3 * i
Explanation
The first step is to find the square root of -48, which is 4√3 * i.
The second step is to multiply 4√3 * i by 2.
So, 2 * (4√3 * i) = 8√3 * i.
Problem 4
What will be the square root of (-24) * 2?
The square root is 4√3 * i
Explanation
To find the square root, calculate (-24) * 2 = -48.
Then, √(-48) = 4√3 * i.
Therefore, the square root of (-24) * 2 is 4√3 * i.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √(-48) units and the width ‘w’ is 6 units.
The perimeter of the rectangle is not a real number.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Length = 4√3 * i,
Width = 6.
Perimeter = 2 × (4√3 * i + 6), which involves adding an imaginary number to a real number, thus it is not a real number.
FAQ on Square Root of -48
1.What is √(-48) in its simplest form?
2.What is the square of -48?
3.Is -48 a perfect square?
4.What are the factors of 48?
5.What is the imaginary unit?
6.How does learning Algebra help students in United States make better decisions in daily life?
7.How can cultural or local activities in United States support learning Algebra topics such as Square Root of -48?
8.How do technology and digital tools in United States support learning Algebra and Square Root of -48?
9.Does learning Algebra support future career opportunities for students in United States?
Important Glossaries for the Square Root of -48
- Imaginary Number: A number that can be written as a real number multiplied by the imaginary unit i, where i is the square root of -1.
- Complex Number: A number composed of a real part and an imaginary part, expressed in the form a + bi, where a and b are real numbers.
- Prime Factorization: The process of expressing a number as the product of its prime factors.
- Square Root: The value that, when multiplied by itself, gives the original number. For negative numbers, it involves the imaginary unit.
- Perfect Square: A non-negative integer that is the square of an integer. Negative numbers cannot be perfect squares.
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Jaskaran Singh Saluja
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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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