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Last updated on May 26th, 2025

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Square Root of -68

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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 various fields, including engineering and physics. Here, we will discuss the square root of -68.

Square Root of -68 for US Students
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What is the Square Root of -68?

The square root is the inverse of the square of a number. Since -68 is a negative number, it does not have a real number as its square root. Instead, the square root of -68 is expressed in terms of imaginary numbers. The square root of -68 is expressed as √-68 = √(68) × √(-1) = 8.2462i, where i is the imaginary unit, defined as √(-1).

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Finding the Square Root of -68

For negative numbers, the square root involves imaginary numbers. The square root of a negative number can be found by separating it into the square root of its positive counterpart and the imaginary unit. Here are the steps to find the square root of -68:

 

  • Separate the negative sign: √(-68) = √(-1) × √(68).
  • Calculate the square root of the positive number: √68 ≈ 8.2462.
  • Combine with the imaginary unit: √-68 = 8.2462i.
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Square Root of -68 by Prime Factorization Method

The prime factorization method is not directly applicable to negative numbers, but it can be used for the positive counterpart of -68. Here is how you can find the prime factors of 68:

 

Step 1: Find the prime factors of 68.

 

Breaking it down, we get 2 × 2 × 17 = 2² × 17.

 

Step 2: Express the square root in terms of prime factors: √68 = √(2² × 17) = 2√17.

 

Since -68 is negative, the square root will involve the imaginary unit: √-68 = 2√17i.

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Square Root of -68 by Long Division Method

The long division method is not applicable for negative numbers when finding their square roots. Instead, we use the long division method for the positive counterpart, 68, and then include the imaginary unit.

 

  • Use the long division method to approximate √68, which is approximately 8.2462.
  • Combine this with the imaginary unit: √-68 = 8.2462i.
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Square Root of -68 by Approximation Method

The approximation method can be used to estimate the square root of the positive part of -68.

 

Step 1: Identify two perfect squares between which 68 lies. The perfect squares are 64 (8²) and 81 (9²).

 

Step 2: Estimate √68 using these bounds. √68 is approximately 8.2462.

 

Step 3: Combine with the imaginary unit to find √-68: √-68 = 8.2462i.

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Common Mistakes and How to Avoid Them in the Square Root of -68

Students often make mistakes while finding the square root of negative numbers, such as disregarding the imaginary unit or improperly handling negative signs. Here are common mistakes and their solutions:

Mistake 1

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Disregarding the Imaginary Unit

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

For example, √-9 should be expressed as 3i, not just 3.

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Square Root of -68 Examples

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Problem 1

Can you find the expression for the area of a square if its side length is √-68 units?

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The area is -68 square units.

Explanation

The area of a square = side².

Given the side length as √-68, the area = (√-68)² = -68.

Since we deal with imaginary numbers, the area is represented as -68 square units in the context of complex numbers.

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Problem 2

A square is designed with an imaginary side of √-68 units. Calculate the perimeter.

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The perimeter is 32.9848i units.

Explanation

Perimeter of a square = 4 × side. Here, side = √-68 = 8.2462i.

Therefore, the perimeter = 4 × 8.2462i = 32.9848i units.

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Problem 3

Multiply √-68 by 3.

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The result is 24.7386i.

Explanation

First, find √-68 = 8.2462i.

Then multiply: 8.2462i × 3 = 24.7386i.

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Problem 4

What is the square of √-68?

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The square is -68.

Explanation

The square of √-68 is (√-68)².

Since √-68 = 8.2462i, then (8.2462i)² = -68.

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Problem 5

If a rectangle has a length of √-68 units and a width of 4 units, what is the area?

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The area is -32.9848 square units.

Explanation

Area of a rectangle = length × width.

Length = √-68 = 8.2462i, width = 4.

Area = 8.2462i × 4 = 32.9848i, expressed as -32.9848 in terms of complex numbers.

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FAQ on Square Root of -68

1.What is √-68 in its simplest form?

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2.How do you handle negative signs in square roots?

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3.What is an imaginary unit?

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4.Why can't we find a real square root of -68?

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5.What is the significance of imaginary numbers?

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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 -68?

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8.How do technology and digital tools in United States support learning Algebra and Square Root of -68?

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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 -68

  • Imaginary Unit: The imaginary unit i is the square root of -1, used to express roots of negative numbers.
     
  • Complex Number: A number that includes both a real part and an imaginary part, such as a + bi.
     
  • Square Root: The value which, when multiplied by itself, gives the original number. For negative numbers, involves the imaginary unit.
     
  • Irrational Number: A number that cannot be expressed as a simple fraction; it has a non-repeating, non-terminating decimal expansion.
     
  • Prime Factorization: The expression of a number as a product of its prime factors, useful for simplifying square roots of positive numbers.
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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 -68, 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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