Problem 3 - Prealgebra

Question

List all square roots of the given number. If the number has no square roots, write “none”. ?289

Step-by-Step Solution

Answer

The square roots of 289 are 17 and -17.

Step 1: Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. So, we are looking for a number \( x \) such that \( x^2 = 289 \).

Step 2: Calculate Positive Square Root

To find the positive square root, calculate \( \sqrt{289} \). This can be done by remembering common square roots: \( \sqrt{289} = 17 \), since \( 17 \times 17 = 289 \).

Step 3: Calculate Negative Square Root

Roots can be positive or negative because converting a negative number squared would also give a positive result. Thus, the negative square root is \( -17 \). This is because \( (-17) \times (-17) = 289 \).

Step 4: Verify Results

Verify that 17 and -17, when squared, both return 289. \( 17^2 = 289 \) and \( (-17)^2 = 289 \). Therefore, both results are correct.

Key Concepts

Positive Square Root

Square roots can sometimes be tricky, but understanding the basics will help you grasp them easily. When we talk about the "positive square root," we're referring to the non-negative number "whose square" results in the given number. In mathematical terms, for a number like 289, the positive square root is 17. Why 17? Because when you multiply 17 by itself, you'd get 289:

• 17 × 17 = 289

It’s straightforward, right? The positive square root symbolized by \( \sqrt{\cdot} \) represents the principal, or primary, root, which is always non-negative. Remembering common squares, like the squares of numbers up to 20, is often helpful when quickly determining positive square roots.

Negative Square Root

The concept of a negative square root might seem perplexing at first, but it's easier than it looks. Just like the positive version, the negative square root gives a square which equals the original number when multiplied by itself. However, it involves a negative counterpart. For our example number 289, besides the positive square root 17, there’s also a negative square root: -17.

• (-17) × (-17) = 289

It works, because multiplying two negative numbers results in a positive product. This is why every positive number actually has two square roots.

Verifying Square Roots

Verifying square roots is a significant step that ensures the accuracy of your results. To verify the square roots of a number, square both the positive and negative roots and check if they return to the original number. Taking our example:

• For the positive square root 17, verify by calculating: \( 17^2 \).
• For the negative square root -17, verify by calculating: \( (-17)^2 \).
• If both calculations result in 289, then they are verified square roots.

Verification is essential because it confirms that your calculated roots are correctly and neatly solving the initial problem. It's like double-checking your work to ensure reliability.