Problem 12 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression evaluates to 4 and -4.

Step 1: Identify the Expression

The expression given is \( \pm \sqrt{16} \), which means that we need to evaluate both the positive and negative square roots of the number 16.

Step 2: Evaluate the Square Root

The square root of 16 is 4 because 4 multiplied by itself (\( 4 \times 4 \)) equals 16. So \( \sqrt{16} = 4 \).

Step 3: Determine the Values

Since the expression involves both positive and negative square roots, the solutions are \( +4 \) and \( -4 \).

Step 4: Final Answer

The expression \( \pm \sqrt{16} \) evaluates to two real numbers: \( 4 \) and \( -4 \).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics that relate to finding a number which, when multiplied by itself, gives the original number. Let's look at the square root of 16 as an example. In this case, the number is 4 because multiplying 4 by itself, i.e., \( 4 \times 4 \), results in 16. When dealing with square roots, especially of perfect squares like 16, it's helpful to remember that they have two possible values: a positive and a negative version. This ties back to the fact that the square of both positive and negative numbers results in a positive number, which is why both \( 4 \) and \( -4 \) are valid solutions. This concept is crucial for understanding real-world problems dealing with symmetry and balance in calculations.

Positive and Negative Values

In mathematics, numbers can be positive or negative, encompassing a whole spectrum of values known as real numbers. They help describe a wide range of quantities, from balances in a bank account to temperatures and altitudes. **Understanding Positive and Negative Signs** - **Positive values**: These numbers are greater than zero and often represent quantities like gains, heights, or credits. - **Negative values**: These numbers are less than zero and may represent losses, depths, or debits. In expressions such as \( \pm \sqrt{16} \), the \( \pm \) symbol indicates the presence of both the positive and the negative values of the square root, which is why \( +4 \) and \( -4 \) are both solutions. Comprehending both signs helps in solving equations and real-life situations where direction and magnitude matter.

Evaluating Expressions

Evaluating mathematical expressions involves calculating them in a way that simplifies or resolves the problem at hand. It's like decoding a language to find the meaning behind numbers and symbols. When given an expression like \( \pm \sqrt{16} \), we start by identifying parts of the expression: the square root and the \( \pm \) symbol. Then, we calculate the square root, which is 4 in this case, and apply both positive and negative signs to get two results: \( +4 \) and \( -4 \).**Steps to Evaluate**:- Identify and interpret each part of the expression.- Calculate any operations, like square roots or exponents.- Apply signs and combine results to find the complete solution. This process helps in understanding equations better and forms a solid foundation for solving more complex mathematical problems. Practicing evaluation improves problem-solving skills and enables us to tackle queries involving real numbers more efficiently.