Question

Evaluate the expression to the nearest hundredth. $$ \frac{-2 \pm 4 \sqrt{2}}{-2} $$

Short answer

The expression evaluates to approximately -1.83 and 3.83.

Step-by-step solution

Evaluate the square root

Following the order of operations, evaluate the square root first. Calculate the square root of 2, which is approximately 1.41.

Perform multiplication

Multiply 4 by the square root of 2 (1.41). It results in approximately 5.65.

Calculate the expression for the positive sign

Evaluate the expression with the positive sign in the numerator first. This gives \(\frac{-2 + 5.65}{-2}\) which is \(-1.83\) approximately.

Calculate the expression for the negative sign

Next, evaluate the expression with the negative sign in the numerator. This gives \(\frac{-2 - 5.65}{-2}\) which is \(3.83\) approximately.

Key concepts

Order of Operations

Understanding the order of operations is fundamental to evaluating algebraic expressions correctly. It's like a set of traffic rules for mathematics that ensures everyone arrives at the same result. The standard order is summarized by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

When you're faced with a complex expression, such as evaluating \( \frac{-2 \pm 4 \sqrt{2}}{-2} \), it's crucial to follow these steps methodically. You would tackle any operations within parentheses first, then address any exponents, which include square roots, and proceed through to multiplication, division, addition, and subtraction. Skipping a step or rearranging them can lead to an incorrect result, just as ignoring traffic signals would cause confusion on the roads.

Square Roots

A square root asks the question: 'What number multiplied by itself gives me the original number?' For instance, the square root of 4 is 2, because 2 times 2 equals 4. Square roots are part of a larger group of operations called radicals, which can sometimes be intimidating, but they follow the same logic as other arithmetic operations.

With square roots, precision is vital since most roots do not result in whole numbers. In our exercise, the square root of 2 is approximately 1.41. This value is then used in subsequent calculations, playing a critical role in reaching the correct answer. So, when dealing with square roots, especially in the context of algebra, an understanding of how to find and use them is essential.

Arithmetic Operations

Arithmetic operations are the bricks and mortar of mathematics. They include addition, subtraction, multiplication, and division. When evaluating algebraic expressions such as \( \frac{-2 \pm 4 \sqrt{2}}{-2} \), each operation has its place and order, as determined by the rules we discussed in the \( \text{Order of Operations} \) section.

These operations have to be applied correctly to compute the given expression. You will always start by resolving any multiplications and divisions before moving on to addition or subtraction. This allows you to simplify the equation step by step, ensuring that each part is handled in isolation, ultimately combining them to finalize the answer.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They may look intimidating at first, but they obey the same basic principles as regular fractions. In our textbook exercise, we're dealing with a rational expression where the numerator includes an expression with a square root and the denominator is a simple integer value.

To evaluate rational expressions, simplify the numerator and denominator as much as possible before dividing. Remember that when dividing by a negative, the signs of the result will invert. Also, when the numerator has an operation like the plus/minus sign, evaluate it separately for each case to determine the different potential outcomes of the expression.