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Last updated on May 26th, 2025

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Square Root of -343

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Understanding the square root of a negative number involves imaginary numbers, as negative numbers do not have real square roots. The concept is applied in various fields, including electrical engineering and control theory. Here, we will discuss the square root of -343.

Square Root of -343 for US Students
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What is the Square Root of -343?

The square root of a negative number involves imaginary numbers. For -343, the square root can be expressed in terms of the imaginary unit 'i'. The square root of -343 is expressed as √(-343) = √(343) × i. Since 343 is a perfect cube, not a perfect square, its square root is irrational. √343 is approximately 18.5203, thus √(-343) = 18.5203i.

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Finding the Square Root of -343

To find the square root of a negative number like -343, we use the concept of imaginary numbers. The real number aspect of the square root is found using normal methods, and the imaginary unit 'i' accounts for the negative sign. Let's understand the following methods:

 

  • Imaginary number concept
  • Approximating the square root of the positive number
  • Using a calculator for complex numbers
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Square Root of 343 by Prime Factorization Method

The prime factorization of 343 helps find its square root. Here’s how 343 is factorized:

 

Step 1: Finding the prime factors of 343 Breaking it down, we get 7 x 7 x 7 or 7³.

 

Step 2: Since 343 is not a perfect square but a perfect cube, we determine it doesn't have a whole number square root. The square root of 343 is expressed as √343 = 18.5203 (approx).

 

Therefore, the square root of -343 is √(-343) = 18.5203i.

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Square Root of 343 by Long Division Method

The long division method is used for non-perfect squares. Here’s how you find √343:

 

Step 1: Pair the digits of 343 from right to left: 43 and 3.

 

Step 2: Find the largest number whose square is less than or equal to 3, which is 1.

 

Step 3: Subtract and bring down the next pair to get 243.

 

Step 4: Double the divisor and find a digit to complete the divisor such that the product is less than or equal to 243. Continue the steps to find √343 ≈ 18.5203.

 

Hence, √(-343) = 18.5203i.

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Square Root of 343 by Approximation Method

Use approximation to find √343, and then apply the imaginary unit:

 

Step 1: Determine the perfect squares near 343. 324 (18²) and 361 (19²).

 

Step 2: Since 343 is between 324 and 361, √343 is between 18 and 19.

 

Step 3: Approximating further using (343 - 324)/(361 - 324) gives a decimal component.

 

Step 4: √343 ≈ 18.5203, thus √(-343) = 18.5203i.

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Common Mistakes and How to Avoid Them in the Square Root of -343

Understanding imaginary numbers is crucial when dealing with the square root of negative numbers. Avoid these common errors:

Mistake 1

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Ignoring the Imaginary Unit

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Always include 'i' for the square root of negative numbers.

For example, √(-49) is 7i, not 7.

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Square Root of -343 Examples

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Problem 1

Can you help Max find the area of a square box if its side length is given as √(-49)?

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The area is not a real number but a complex number.

Explanation

The side length is √(-49) = 7i.

The area is side² = (7i)² = 49(-1) = -49.

Thus, the area is -49, indicating it's not real.

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Problem 2

A square-shaped building measuring 343 square feet is constructed; if each side is √(-343), what will be the area of half the building?

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It is not possible to have a real measurement for half the area.

Explanation

Since √(-343) is not real, the area calculation results in a complex number, making it impractical for real-world application.

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Problem 3

Calculate √(-343) x 3.

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55.5609i

Explanation

First, find √(-343) = 18.5203i.

Then multiply by 3: 18.5203i x 3 = 55.5609i.

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Problem 4

What is the square root of (-324 + 1)?

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18i

Explanation

Simplify (-324 + 1) = -323.

The square root is √(-323) = √323 × i.

Approximating √323 ≈ 17.972, so √(-323) ≈ 17.972i.

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Problem 5

Find the perimeter of the rectangle if its length 'l' is √(-343) units and the width 'w' is 38 units.

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The perimeter is a complex number: 76 + 37.0406i.

Explanation

Perimeter = 2 × (length + width).

Length = 18.5203i, so perimeter = 2 × (18.5203i + 38) = 76 + 37.0406i.

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FAQ on Square Root of -343

1.What is √(-343) in its simplest form?

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2.What are the prime factors of 343?

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3.Calculate the square of 343.

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4.Is 343 a prime number?

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5.What is the real-part square root of 343?

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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 -343?

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8.How do technology and digital tools in United States support learning Algebra and Square Root of -343?

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

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Important Glossaries for the Square Root of -343

  • Imaginary number: A number that when squared gives a negative result, often expressed with 'i', where i² = -1.

 

  • Complex number: A number that has both a real and an imaginary part, such as a + bi.

 

  • Cube root: The number that when multiplied by itself three times gives the original number, e.g., ∛343 = 7.

 

  • Prime factorization: Breaking down a number into its prime factors, e.g., 343 = 7 x 7 x 7.

 

  • Perfect square: A number that is the square of an integer, e.g., 16 is a perfect square (4²).
Professor Greenline from BrightChamps

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 -343, 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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