Problem 13 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ -\sqrt{16} $$

Step-by-Step Solution

Answer

The value of \\(-\sqrt{16}\\) is \\(-4\\).

Step 1: Identify the Square Root

First, identify the square root of the number under the radical. In this case, you need to find the square root of 16, which is 4 since 4 multiplied by itself gives 16.

Step 2: Apply the Negative Sign

Use the negative sign outside of the square root to change the positive square root obtained in the previous step to negative. So, applying the negative sign to 4 gives \(-4\).

Key Concepts

Understanding the Square Root

A square root is a value that, when multiplied by itself, results in the original number. In the expression \(-\sqrt{16}\), you first need to find the square root of 16. When you say "What is the square root of 16?", you are actually asking "What number times itself equals 16?". The answer is 4 because \(4 \times 4 = 16\). Understanding this concept is fundamental as it is often used in various mathematical expressions and problem-solving scenarios.

• The square root symbol \(\sqrt{}\) is like a cue to take the value inside, and find what number squared (or multiplied by itself) makes that value.
• Square roots often have two solutions: a positive and a negative. However, the context usually determines which one we use.
• In this expression, we're only focused on the positive square root initially, which is 4.

The Role of the Negative Sign

In expressions like \(-\sqrt{16}\), the negative sign plays a crucial role. It appears outside the square root, indicating that the final result should be negative.
When evaluating an expression like this, you first calculate the square root as a positive number. For \(\sqrt{16}\), this number is 4. Then, you apply the negative sign from the beginning of the expression. The negative sign switches the sign of the root result from positive to negative. Thus, you end up with \(-4\).

• Think of the negative sign as a "flipping" tool - it changes a positive result to a negative one.
• It's not altering the operation you perform, just the sign of the result.
• Always evaluate the square root part first before considering any signs outside the radical.

Expression Evaluation Explained

When you evaluate expressions involving square roots and negative signs, understanding the order of operations is key to getting the correct result. In mathematics, operations are typically performed in a specific order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). However, with expressions like \(-\sqrt{16}\), the steps simplify to focusing mostly on the square root and the handling of the negative sign.
Here’s how you evaluate such expressions:

• First, tackle the square root to find its value - in this case, 4.
• Then, apply any external operations, like applying a negative sign, as demonstrated here.

Evaluation ensures each component is accurately calculated before applying operations outside the main expression.