Problem 14 - Algebra 2

Question

Simplify. $$ \pm \sqrt{169} $$

Step-by-Step Solution

Answer

Simplified, it's \( \pm 13 \).

Step 1: Identify the Expression

The expression given is \( \pm \sqrt{169} \). The symbol \( \pm \) indicates both the positive and negative square roots.

Step 2: Calculate the Square Root

Calculate \( \sqrt{169} \). The square root of 169 is 13 because \( 13 \times 13 = 169 \).

Step 3: Apply the Plus-Minus Sign

Since the expression includes \( \pm \), include both positive and negative results of the square root. Therefore, \( \pm \sqrt{169} = \pm 13 \), which means 13 and -13.

Key Concepts

Understanding Square Roots

Square roots are a fundamental concept in mathematics. When you see a square root symbol, also known as a radical (\( \sqrt{} \)), it means you need to find a number that, when multiplied by itself, gives you the original number under the radical. For example, the square root of 169 is written as \( \sqrt{169} \). To find this, consider what number times itself equals 169. Here, 13 times 13 equals 169, so \( \sqrt{169} = 13 \).
Potentially, numbers like 4, 16, and 25 also have square roots because they are perfect squares (e.g., the square root of 16 is 4). Understanding this concept allows you to simplify mathematical expressions efficiently. Square roots can apply to both positive and negative numbers, provided you account for the suitable context or symbols used.

Positive and Negative Numbers Explained

In dealing with expressions such as \( \pm \sqrt{169} \), understanding positive and negative numbers is crucial. Positive numbers are any numbers greater than zero. They can be whole numbers, fractions, or decimals, indicated by the absence of a negative sign or after a plus sign. Examples include 13, 5.7, and 3/4.
On the flip side, negative numbers are less than zero and carry a negative sign or a minus sign. Examples include -13, -5.7, and -3/4. Negative numbers are just as important in mathematical expressions as positive numbers.
In expressions like this, you can have both the positive and negative result of a calculation. For \( \pm \sqrt{169} \), this means including both 13 (positive) and -13 (negative). This ensures you've fully covered all possibilities from the square root while respecting mathematical rules.

Exploring Mathematical Symbols

Mathematical symbols are shorthand ways of expressing ideas and operations in math quickly and accurately. Let's focus on some essential symbols.
The square root symbol (\( \sqrt{} \)) shows that you want the square root of a number. When used in an expression like \( \pm \sqrt{169} \), it means find the square root without specifying positive or negative.
The plus-minus symbol (\( \pm \)) is unique because it indicates two values simultaneously. It tells us to consider both positive and negative results of an operation. For example, with \( \pm 13 \), we accept both 13 and -13.

• Equals Sign (\( = \)): Shows equality between two expressions.
• Multiplication Sign (\( \times \) or \( \cdot \)): Indicates the operation of multiplying.
• Addition and Subtraction (\( + \) and \( - \)): Basic arithmetic operations to find the sum or difference of numbers.

Symbols in mathematics are more than just characters; they provide structure and clarity to mathematical communication.