Square Root of -68
2026-02-28 09:47 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-minus-68

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Last updated on August 5, 2025

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.

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).

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.

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.

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.

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:

Problem 1

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

Okay, lets begin

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.

Okay, lets begin

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.

Well explained 👍

Problem 3

Multiply √-68 by 3.

Okay, lets begin

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?

Okay, lets begin

The square is -68.

Explanation

The square of √-68 is (√-68)².

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

Well explained 👍

Problem 5

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

Okay, lets begin

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.

Well explained 👍

FAQ on Square Root of -68

1.What is √-68 in its simplest form?

The simplest form of √-68 is 8.2462i, where i is the imaginary unit.

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

For negative numbers, separate the negative sign and use the imaginary unit: √(-x) = √x × i.

3.What is an imaginary unit?

The imaginary unit, denoted as i, is defined as √(-1). It is used to express square roots of negative numbers.

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

Real numbers squared are always non-negative. Therefore, negative numbers do not have real square roots, only imaginary ones.

5.What is the significance of imaginary numbers?

Imaginary numbers are crucial in complex number systems, allowing solutions to equations that have no real solutions.

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