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Last updated on May 26th, 2025

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Square Root of -288

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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 square root is used in the field of vehicle design, finance, etc. Here, we will discuss the square root of -288.

Square Root of -288 for US Students
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What is the Square Root of -288?

The square root is the inverse of the square of the number. Since -288 is a negative number, its square root is not a real number. The square root of -288 is expressed in terms of the imaginary unit 'i'. In radical form, it is expressed as √(-288) = √288 × i, while in exponential form, it is (288)^(1/2) × i. The square root of 288 is 16.97056, so the square root of -288 is 16.97056i, which is an imaginary number.

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Finding the Square Root of -288

The prime factorization method is used for perfect square numbers. However, since -288 is not a perfect square and is negative, we focus on finding the square root of 288 first, then multiply by 'i'. Let's learn the following methods:

 

  • Prime factorization method
  • Long division method
  • Approximation method
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Square Root of -288 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now, let us look at how 288 is broken down into its prime factors:

 

Step 1: Finding the prime factors of 288 Breaking it down, we get 2 × 2 × 2 × 2 × 3 × 3: \(2^4 \times 3^2\)

 

Step 2: Now, we found out the prime factors of 288. The second step is to make pairs of those prime factors. Since 288 is not a perfect square, the digits of the number can’t be grouped perfectly into pairs. Therefore, calculate √288 using prime factorization: \(2^2 \times 3\) yields 12√2.

 

Step 3: The square root of -288 is thus √288 × i = 12√2 × i.

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Square Root of -288 by Long Division Method

The long division method is particularly used for non-perfect square numbers. In this method, we check the closest perfect square number for the given number and then multiply by 'i'. Let's learn how to find the square root step by step:

 

Step 1: Group the numbers from right to left for 288, which is grouped as 88 and 2.

 

Step 2: Find n whose square is 2. We use 1 because 1 × 1 is less than or equal to 2. The quotient is 1, and after subtracting 1 from 2, the remainder is 1.

 

Step 3: Bring down 88, making the new dividend 188. Add the old divisor with itself, 1 + 1, to get 2 as the new divisor.

 

Step 4: Find 2n × n ≤ 188. Let n be 7, because 27 × 7 = 189.

 

Step 5: Subtract 188 from 189 to get a remainder of 1.

 

Step 6: Since we have a remainder, we continue the division process. The quotient becomes 16.97056, so the square root of 288 is approximately 16.97056.

 

Step 7: Since we are dealing with -288, the square root of -288 is 16.97056i.

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Square Root of -288 by Approximation Method

Approximation is another method for finding square roots, which is easy for estimating the square root of a given number, then multiplying by 'i'. Now, let us learn how to find the square root of -288 using approximation:

 

Step 1: Find the closest perfect square of √288. The closest perfect squares to 288 are 256 and 289. √288 falls between 16 and 17.

 

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (288 - 256) / (289 - 256) = 0.97.

 

Step 3: Add this decimal to the integer square root. 16 + 0.97 = 16.97. Step 4: The square root of -288 is 16.97i.

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Common Mistakes and How to Avoid Them in the Square Root of -288

Students often make mistakes while finding square roots, such as forgetting the imaginary unit when dealing with negative numbers. Let's look at a few common mistakes in detail.

Mistake 1

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Forgetting about the Imaginary Unit

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It is essential to remember that the square root of a negative number involves the imaginary unit 'i'.

For example, √(-50) = √50 × i = 7.071067i.

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Square Root of -288 Examples

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Problem 1

If the side length of a square is √(-72), can you find the area?

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The area is -72 square units.

Explanation

The area of the square = side².

The side length is given as √(-72).

Area of the square = (√(-72))² = (√72 × i)² = 72 × (-1) = -72.

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Problem 2

A square-shaped space has an area of -288 square feet. What is the length of each side?

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16.97i feet.

Explanation

The side length of the square is the square root of the area.

Since the area is -288, the side length is √(-288) = 16.97i feet.

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Problem 3

Calculate √(-288) × 5.

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84.85i

Explanation

First, find the square root of -288, which is 16.97i.

Then, multiply 16.97i by 5.

So, 16.97i × 5 = 84.85i.

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Problem 4

What is the square root of (-72 + 72)?

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0

Explanation

To find the square root, calculate the sum (-72 + 72) = 0.

The square root of 0 is 0.

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Problem 5

Find the perimeter of a rectangle with length 'l' as √(-72) units and width 'w' as 10 units.

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The perimeter is 20 + 16.97i units.

Explanation

Perimeter of the rectangle = 2 × (length + width)

Perimeter = 2 × (√(-72) + 10) = 2 × (8.485i + 10) = 20 + 16.97i units.

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FAQ on Square Root of -288

1.What is √(-288) in its simplest form?

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2.Is -288 a perfect square?

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3.What is the square of -288?

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4.Can the square root of a negative number be real?

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5.What is the imaginary unit 'i'?

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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 -288?

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8.How do technology and digital tools in United States support learning Algebra and Square Root of -288?

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

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Important Glossary for the Square Root of -288

  • Imaginary Number: A number that can be expressed as a real number multiplied by the imaginary unit 'i', where i² = -1.

 

  • Square Root: The square root of a number is a value that, when multiplied by itself, gives the original number.

 

  • Imaginary Unit (i): The imaginary unit 'i' is defined as √(-1), used to represent the square roots of negative numbers.

 

  • Prime Factorization: The expression of a number as a product of its prime factors.

 

  • Approximation Method: A method used to estimate the value of a square root for non-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 -288, 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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Fun Fact

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

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