BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon102 Learners

Last updated on May 26th, 2025

Math Whiteboard Illustration

Square Root of -289

Professor Greenline Explaining Math Concepts

The square of a number is the result of multiplying the number by itself. The inverse operation is finding the square root. Square roots are essential in various fields, including complex number analysis and engineering. Here, we will discuss the square root of -289.

Square Root of -289 for Vietnamese Students
Professor Greenline from BrightChamps

What is the Square Root of -289?

The square root is the inverse operation of squaring a number. Since -289 is negative, its square root is not a real number. Instead, it is expressed using the imaginary unit. The square root of -289 can be written as √-289, which simplifies to 17i, where i is the imaginary unit defined by i² = -1.

Professor Greenline from BrightChamps

Understanding the Square Root of -289

To understand the square root of a negative number, we use the concept of imaginary numbers. The imaginary unit i is defined such that i² = -1. Therefore, the square root of -289 is computed using the relationship: √-289 = √(289 × -1) = √289 × √-1 = 17i.

Professor Greenline from BrightChamps

Properties of Square Roots of Negative Numbers

The properties of square roots of negative numbers rely on the imaginary unit i. Key points include:

 

1. The square root of a negative number always involves the imaginary unit i.

 

2. (√-a)² = -a for any positive real number a.

 

3. Imaginary numbers are used in complex number calculations, where a complex number is of the form a + bi.

Professor Greenline from BrightChamps

Applications of Imaginary Numbers

Imaginary numbers have applications in various fields:

 

1. Engineering: Used in signal processing and control systems.

 

2. Physics: Appear in quantum mechanics and electrical engineering.

 

3. Mathematics: Essential for solving certain polynomial equations.

 

4. Computer Science: Used in algorithms dealing with complex numbers.

Professor Greenline from BrightChamps

Common Mistakes with Square Roots of Negative Numbers

When dealing with square roots of negative numbers, common mistakes include:

 

1. Assuming the square root of a negative number is real.

 

2. Forgetting the imaginary unit i in calculations.

 

3. Misinterpreting i² = -1 as a real number operation.

Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in the Square Root of -289

Students often make mistakes when working with square roots of negative numbers. It's important to correctly apply the concept of imaginary numbers. Let's explore some common errors and how to avoid them.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Ignoring the Imaginary Component

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

A frequent mistake is treating the square root of a negative number as if it were real.

The square root of -289 is not real; it is 17i.

Always include the imaginary unit i for negative square roots.

Max from BrightChamps Saying "Hey"

Square Root of -289 Examples

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

What is the product of the square root of -289 and 3?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The product is 51i.

Explanation

The square root of -289 is 17i.

Multiply this by 3: 17i × 3 = 51i.

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 2

If z = 5 + √-289, what is the conjugate of z?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The conjugate of z is 5 - 17i.

Explanation

The conjugate of a complex number a + bi is a - bi.

Given z = 5 + 17i, its conjugate is 5 - 17i.

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 3

How do you express the square of the square root of -289?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The expression is -289.

Explanation

The square of the square root of -289 is (√-289)² = (17i)² = 17² × i² = 289 × (-1) = -289.

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 4

What is the imaginary part of the square root of -289?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

The imaginary part is 17.

Explanation

The square root of -289 is 17i.

The imaginary part is the coefficient of i, which is 17.

Max from BrightChamps Praising Clear Math Explanations
Max, the Girl Character from BrightChamps

Problem 5

If the imaginary unit is defined as i, what is i raised to the power of 4?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"

i raised to the power of 4 is 1.

Explanation

i is defined such that i² = -1.

Therefore, i⁴ = (i²)² = (-1)² = 1.

Max from BrightChamps Praising Clear Math Explanations
Ray Thinking Deeply About Math Problems

FAQ on Square Root of -289

1.What is √-289 in terms of imaginary numbers?

Math FAQ Answers Dropdown Arrow

2.Is -289 a perfect square?

Math FAQ Answers Dropdown Arrow

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

Math FAQ Answers Dropdown Arrow

4.What are the applications of imaginary numbers?

Math FAQ Answers Dropdown Arrow

5.How do you express a negative square root in terms of i?

Math FAQ Answers Dropdown Arrow

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

Math FAQ Answers Dropdown Arrow

7.How can cultural or local activities in Vietnam support learning Algebra topics such as Square Root of -289?

Math FAQ Answers Dropdown Arrow

8.How do technology and digital tools in Vietnam support learning Algebra and Square Root of -289?

Math FAQ Answers Dropdown Arrow

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

Math FAQ Answers Dropdown Arrow
Professor Greenline from BrightChamps

Important Glossaries for the Square Root of -289

  • Imaginary unit (i): A mathematical concept where i is defined such that i² = -1. It is used to express the square roots of negative numbers.

 

  • Complex number: A number in the form a + bi, where a and b are real numbers, and i is the imaginary unit.

 

  • Square root: The inverse operation of squaring a number. For a number x, the square root is a number y such that y² = x.

 

  • Conjugate: The conjugate of a complex number a + bi is a - bi. It is used in complex number operations.

 

  • Perfect square: A number that is the square of an integer. Negative numbers are not perfect squares in the real number system, but their roots are expressed using imaginary numbers.
Professor Greenline from BrightChamps

About BrightChamps in Vietnam

At BrightChamps, we know algebra is more than symbols—it’s a path to countless opportunities! Our goal is to help children across Vietnam grasp essential math skills, with today’s focus on the Square Root of -289 and a special look at square roots—in an engaging, enjoyable, and easy-to-learn way. Whether your child is figuring out how fast a roller coaster moves at Suoi Tien Theme Park, keeping track of local football scores, or budgeting their allowance for new gadgets, mastering algebra gives them the confidence to handle daily challenges. Our interactive lessons make learning easy and fun. Since children in Vietnam learn in different ways, we adapt to each learner’s style. From Ho Chi Minh City’s vibrant streets to the beautiful Ha Long Bay, BrightChamps makes math come alive throughout Vietnam. Let’s make square roots an exciting part of every child’s math adventure!
Math Teacher Background Image
Math Teacher Image

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.

Math Teacher Fun Facts Image
Max, the Girl Character from BrightChamps

Fun Fact

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

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
Dubai - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom