Problem 37 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Check the results by squaring each root. $$ -\sqrt{1} $$

Step-by-Step Solution

Answer

The evaluation of the expression \( -\sqrt{1} \) is \( -1 \) and the result can be checked by squaring -1 which gives 1.

Step 1: Evaluating the Square Root

Firstly, the square root of 1 is calculated. The square root of 1 is 1, because \(1 \times 1 = 1 \).

Step 2: Applying the Negative Sign

Next, the given expression holds a negative sign before the square root. So the complete evaluation would be the negative square root of 1. Therefore, the evaluated expression is \( -1 \) because it was \( - \) before the square root.

Step 3: Verifying the Result by Squaring

Lastly, to check the result, square the evaluated result, i.e., \( (-1)^2 \). Squaring -1 gives us 1, which verifies our initial operation.

Key Concepts

Algebra

In algebra, square roots are a fundamental concept. When you see a symbol like \( \sqrt{} \), it's asking you to find a number which, when multiplied by itself, gives the original number inside the root. For instance, \( \sqrt{1} = 1 \) because \( 1 \times 1 = 1 \). However, lengthy algebraic expressions involving square roots often include signs that affect the outcome of the expression, such as a negative sign.

• The square root function will only give the principal (non-negative) root in the absence of additional details.
• When a minus sign is placed outside the square root, it is applied to the result of the square root, changing its sign.

This blending of operations is where algebra comes in handy. It involves structuring and simplifying expressions by using rules of operation in a systematic way. By organizing expressions correctly, you can easily apply further mathematical operations.

Negative Numbers

Working with negative numbers can be tricky, especially when they intersect with other operations like square roots. A negative number is any number less than zero. When you see a negative sign before a square root, it indicates that the final result of the square root should be multiplied by -1.

• For instance, \( -\sqrt{1} \) is evaluated by first finding \( \sqrt{1} \), which is 1, and then applying the negative sign to make it \( -1 \).
• Operations with negative numbers follow unique rules, particularly when they are multiplied or squared.

A common rule is remembering that multiplying two negatives results in a positive, while multiplying a negative by a positive results in a negative. Understanding these rules helps prevent errors in calculations involving negative numbers and square roots.

Verification by Squaring

Verification by squaring is a useful tool to confirm if your solution is correct when working with square roots. After calculating a square root expression, you can square your result to ensure it matches the original value under the square root.

• Once the square root is evaluated, like \( -1 \) from \( -\sqrt{1} \), you can check your work by squaring it: \( (-1)^2 = 1 \).
• The squaring process reverses the square root operation, returning you to the original number before the square root was applied.

This step is crucial because it helps verify the accuracy of operations which can become error-prone due to signs. It acts as a double-check mechanism, offering a safeguard against missteps involved in the calculation.