What Is The Root Of 289

What is the Square Root of 289? Finding the Perfect Solution

Finding the square root of a number is a fundamental concept in mathematics. This article will explore how to calculate the square root of 289, providing a clear explanation of the process and offering some helpful tips for similar calculations. Understanding this will enhance your problem-solving skills in algebra and beyond.

The square root of a number is a value that, when multiplied by itself, gives the original number. In simpler terms, it's the number that, when squared, equals the target number. So, we're looking for a number x such that x² = 289.

There are several ways to approach this:

Method 1: Using Prime Factorization

This method is particularly useful for larger numbers. We break down 289 into its prime factors:

1. Start with the smallest prime number, 2: 289 is not divisible by 2.
2. Try the next prime number, 3: 289 is not divisible by 3.
3. Try 5: 289 is not divisible by 5.
4. Try 7: 289 is not divisible by 7.
5. Try 11: 289 is not divisible by 11.
6. Try 13: 289 / 13 = 22.23... It's not divisible by 13 either.
7. Try 17: 289 / 17 = 17. This is our breakthrough!

Therefore, 289 can be factored as 17 x 17, or 17². Since 17 multiplied by itself equals 289, the square root of 289 is 17.

Method 2: Using a Calculator

The simplest way to find the square root of 289 is by using a calculator. Most calculators have a square root function (often denoted by √ or sqrt). Simply input 289 and press the square root button. The result will be 17.

Method 3: Estimation and Trial and Error

If you don't have a calculator handy, you can estimate. Knowing that 10² = 100 and 20² = 400, we know the square root of 289 lies somewhere between 10 and 20. You can then systematically test numbers within this range:

• 15² = 225
• 16² = 256
• 17² = 289

This method, while time-consuming for larger numbers, demonstrates the underlying principle of square roots.

Conclusion: The Square Root of 289 is 17

Regardless of the method used, the square root of 289 is unequivocally 17. Understanding the different approaches allows you to solve similar problems efficiently, whether you're working with a calculator, relying on prime factorization, or employing estimation techniques. Remember, practicing these methods will strengthen your mathematical skills and problem-solving abilities.