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Last updated on May 26th, 2025

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Square Root of -48

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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.

Square Root of -48 for US Students
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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).

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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.

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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.

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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.

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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.

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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

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Forgetting the Imaginary Unit

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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.

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Square Root of -48 Examples

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Max, the Girl Character from BrightChamps

Problem 1

Can you help Max find the area of a square box if its side length is given as √(-48)?

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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.

 

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Max, the Girl Character from BrightChamps

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?

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-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.

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Max, the Girl Character from BrightChamps

Problem 3

Calculate 2 * √(-48).

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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.

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Max, the Girl Character from BrightChamps

Problem 4

What will be the square root of (-24) * 2?

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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.

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Max, the Girl Character from BrightChamps

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √(-48) units and the width ‘w’ is 6 units.

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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.

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FAQ on Square Root of -48

1.What is √(-48) in its simplest form?

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2.What is the square of -48?

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3.Is -48 a perfect square?

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4.What are the factors of 48?

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5.What is the imaginary unit?

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6.How does learning Algebra help students in United States make better decisions in daily life?

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7.How can cultural or local activities in United States support learning Algebra topics such as Square Root of -48?

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8.How do technology and digital tools in United States support learning Algebra and Square Root of -48?

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9.Does learning Algebra support future career opportunities for students in United States?

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Professor Greenline from BrightChamps

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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About BrightChamps in United States

At BrightChamps, we understand algebra is more than just symbols—it’s a gateway to endless possibilities! Our goal is to empower kids throughout the United States to master key math skills, like today’s topic on the Square Root of -48, with a special emphasis on understanding square roots—in an engaging, fun, and easy-to-grasp manner. Whether your child is calculating how fast a roller coaster zooms through Disney World, keeping track of scores during a Little League game, or budgeting their allowance for the latest gadgets, mastering algebra boosts their confidence to tackle everyday problems. Our hands-on lessons make learning both accessible and exciting. Since kids in the USA learn in diverse ways, we customize our methods to suit each learner’s style. From the lively streets of New York City to the sunny beaches of California, BrightChamps brings math alive, making it meaningful and enjoyable all across America. Let’s make square roots an exciting part of every child’s math adventure!
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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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Max, the Girl Character from BrightChamps

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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