Problem 2 - Prealgebra

Question

List all square roots of the given number. If the number has no square roots, write “none”. 361

Step-by-Step Solution

Answer

The square roots of 361 are 19 and -19.

Step 1: Identify the Given Number

The given number is 361. Our task is to find its square roots.

Step 2: Calculate the Positive Square Root

To find the square root of 361, we need to determine what number, when multiplied by itself, equals 361. We will compute the positive square root first.\[\sqrt{361} = 19\]

Step 3: Consider the Negative Square Root

Square roots can be both positive and negative. Thus, we must also consider the negative version of the square root found in Step 2.\[-\sqrt{361} = -19\]

Step 4: List All Square Roots

Both the positive and negative values calculated are square roots of 361. Therefore, the square roots of 361 are 19 and -19.

Key Concepts

Positive Square Root

The positive square root of a number is the value that, when multiplied by itself, gives the original number. It's symbolized as \( \sqrt{n} \) where \( n \) is the number in question. For example, when you have \( \sqrt{361} \), you are seeking the positive number that, when squared (multiplied by itself), equals 361. In this case, the positive square root of 361 is 19.

Here’s a quick way to verify this: calculate 19 multiplied by 19. It equals 361, confirming that 19 is indeed the positive square root. The positive square root is often what we are most interested in, especially when discussing things like the length of a side of a square, since lengths are always positive.

Points to remember:

• The positive square root is represented without a negative sign.
• It’s the principal root, typically referred to unless specified otherwise.
• It's commonly used in real-world measurements where negative values aren't plausible.

Negative Square Root

The negative square root of a number might not always come to mind immediately, but it's just as important. It is simply the negative version of the positive square root. Symbolically, it’s written as \(-\sqrt{n}\). For the number 361, the negative square root is \(-19\) because \(-19 \times -19\) also equals 361. The product of two negative numbers is positive, which is why \(-19\) squared results in 361.

In many mathematical situations, recognizing negative square roots is essential. For instance, when solving quadratic equations, negative roots play a vital role in conveying all possible solutions.

Key insights on negative square roots:

• They accompany positive square roots to form a complete set of solutions for equations involving squaring.
• Not usually used in practical situations unless dealing with purely mathematical concepts.

Square Root Properties

Understanding the properties of square roots helps you manage and solve equations more effectively. Here are some important properties that relate to both positive and negative square roots:

• Square roots simplify: The square root of a product \( \sqrt{a \times b} \) can be simplified to \( \sqrt{a} \times \sqrt{b} \), if \( a \) and \( b \) are non-negative, simplifying calculations significantly.
• Product of opposite roots: The product of the positive and negative square root of the same number always results in that number being negated. For instance, \((19) \times (-19) = -361\).
• Positive and negative roots: They indicate symmetry in mathematics; for any positive square root, there is a corresponding negative square root, indicating symmetry in solutions.

Being familiar with these properties allows you to manipulate expressions for better understanding, simplify complex problems, and appreciate the duality and balance present in mathematics.