Maths

Square Root of 289 – Definition, Solution & Examples

Square Root of 289 – Definition, Solution & Examples

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How to Find the Square Root of 289 Step by Step

The concept of square root of 289 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.

What Is Square Root of 289?

A square root of 289 is defined as the number which, when multiplied by itself, gives 289. In mathematical terms, \( \sqrt{289} = 17 \) because \( 17 \times 17 = 289 \). You’ll find this concept applied in topics such as perfect squares, area calculation, and solving quadratic equations.

Key Formula for Square Root of 289

Here’s the standard formula: \( \sqrt{289} = 17 \)

Cross-Disciplinary Usage

Square root of 289 is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in geometry, algebra, and data analysis questions.

Step-by-Step Illustration

Start with the question: What is \( \sqrt{289} \)?

Since 289 is a perfect square, find which number squared gives 289.
Prime Factorization Method:

1. Write 289 as 17 × 17 (since 17 is a prime number).

2. Now, \( \sqrt{17 \times 17} = 17 \).
Long Division Method:

1. Group 289 as “2” and “89”.

2. 4 × 4 = 16 (try 1 digit). Next, bring down 89.

3. Through steps, you reach 17 as the answer.
Final Answer:

The square root of 289 is 17.

Speed Trick or Vedic Shortcut

Here’s a quick shortcut that helps solve problems faster when working with square root of 289. Many students use this trick during timed exams to save crucial seconds.

Example Trick: If you know 289 ends with 9 and is near 16² (256) and 17² (289), guess which integer square is closest. Since 17 × 17 = 289, the answer is 17.

Recognize that perfect squares from 10² (100) to 20² (400) are easy to memorize for exams.
Spot the last two digits: 89 in 289 match the second digit pattern of some common squares (e.g., 17² = 289).
Try squaring 17 quickly: \( 17 \times 17 = 17 \times (10 + 7) = 170 + 119 = 289 \).
Final Answer: 17 is the square root of 289.

Tricks like this aren’t just cool — they’re practical in competitive exams like NTSE, Olympiads, and even JEE. Vedantu’s live sessions include more such shortcuts to help you build speed and accuracy.

Try These Yourself

Find the square root of 28900.
Check if 361 is a perfect square like 289.
Find all perfect square numbers between 250 and 300.
Solve: What is the square root of 225?

Frequent Errors and Misunderstandings

Confusing square root of 289 (\( \sqrt{289} \)) with “half of 289.”
Assuming 289 is not a perfect square due to last digit pattern.
Skipping stepwise calculation and making errors in long division.

Relation to Other Concepts

The idea of square root of 289 connects closely with topics such as Square Root of 169 and Square Root of 225. Mastering this helps with understanding more advanced concepts in algebra and geometry.

Classroom Tip

A quick way to remember the square root of 289 is that it matches 17 squared: 17 × 17 = 289. Vedantu’s teachers often use this multiplication trick as a visual cues chart for perfect squares up to 20 × 20 during live classes for rapid recall.

Wrapping It All Up

We explored square root of 289—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept.

FAQs on Square Root of 289 – Definition, Solution & Examples

1. What is the square root of 289?

The square root of 289 is 17, because 17 multiplied by itself (17 x 17) equals 289. This means 289 is a perfect square.

2. How do I find the square root of 289 using the prime factorization method?

First, find the prime factors of 289. 289 breaks down into 17 x 17. Since there's a pair of 17s, the square root is 17.

3. How do I calculate the square root of 289 using the long division method?

The long division method is a more complex process for finding square roots. However, since 289 is a perfect square, you would find that the process would lead to the answer 17.

4. Is the square root of 289 rational or irrational?

The square root of 289 (which is 17) is a rational number because it can be expressed as a fraction (17/1).

5. What are the factors of 289?

The factors of 289 are 1, 17, and 289.

6. What is the square root of 28900?

The square root of 28900 is 170, because 170 x 170 = 28900.

7. Can the square root of 289 be simplified further?

No, 17 is the simplest form of the square root of 289. It's already a whole number.

8. Is √289 a real number?

Yes, √289 (which equals 17) is a real number. Real numbers include all rational and irrational numbers.

9. How is finding the square root of 289 useful in solving quadratic equations?

Understanding square roots is crucial for solving quadratic equations. For example, if you have x² = 289, then x = ±√289 = ±17.

10. Are there any real-world applications of finding the square root of 289 (or similar calculations)?

Square roots are used extensively in various fields like geometry (calculating areas or lengths), physics (solving equations involving distances or velocities), and engineering (designing structures or circuits). For example, if a square has an area of 289 square units, its side length would be 17 units (√289).

11. What is the difference between the principal square root and other square roots?

The principal square root is the non-negative square root of a number. While both 17 and -17 squared equal 289, the principal square root is 17.

12. How can knowing perfect squares help me solve problems faster?

Recognizing perfect squares like 289 (17²) allows for quicker calculations and problem-solving. It helps you spot patterns and simplify expressions efficiently.