Problem 86 - Elementary Algebra Within Reach

Question

Evaluate the expression. $$\frac{-\left|7+3^{2}\right|}{4}$$

Step-by-Step Solution

Answer

The value of the expression \(\frac{-|7+3^{2}|}{4}\) is -4.

Step 1: Evaluate the expression in the absolute value sign

Calculate the expression inside the absolute value, which is \(7 + 3^{2}\). The exponent takes priority according to the order of operations (PEMDAS), so first square 3 to get 9, and then add 7 to this result, giving a result of 16.

Step 2: Apply the absolute value operation

The absolute value of 16 is 16, because 16 is a positive number and the absolute value of any positive number is the number itself.

Step 3: Divide by 4

Divide the number you got from taking the absolute value, by 4. This gives \(\frac{-16}{4} = -4\).

Key Concepts

Order of Operations

When evaluating mathematical expressions, it's essential to follow a specific sequence called the Order of Operations. This ensures that everyone interprets and solves expressions consistently and accurately. The Order of Operations can be remembered using the acronym PEMDAS:

• Parentheses - Solve expressions inside parentheses or brackets first.
• Exponents - Evaluate powers or roots next.
• Multiplication and Division - Conduct these operations from left to right.
• Addition and Subtraction - Perform these last, also from left to right.

In the exercise given, following the Order of Operations means evaluating the exponent first. The expression inside the absolute value is \(7 + 3^2\). Here, \(3^2\) must be evaluated first, giving us 9. Then, you add 7, resulting in 16.

Absolute Value

Absolute value is a fundamental concept in mathematics that represents the distance of a number from zero on the number line. No matter the number sign (positive or negative), the absolute value is always a positive number or zero. In other words, absolute value transforms negative numbers into their positive counterparts while leaving positive numbers unchanged.
For the expression \(|7 + 3^2|\), once inside the absolute value, becomes \(|16|\). Since 16 is already positive, the absolute value keeps it 16.
The absolute value is important as it's widely used in both arithmetic calculations and more advanced mathematics. It ensures that when calculating distances or magnitudes, we only get non-negative results.

Simplifying Expressions

Simplifying expressions is about making mathematical expressions easier to work with by consolidating numbers and variables. The ultimate aim is to end up with the simplest form possible without changing the expression's value.
In our exercise:

• After evaluating \(|7 + 3^2|\) to obtain 16, this value substitutes into the expression.
• Next, the expression becomes \(\frac{-16}{4}\).
• Simplifying this fraction yields -4.

Simplifying is crucial because it provides a cleaner, more understandable result. This makes it easier to interpret the meaning behind the numbers in any context or problem. Throughout mathematics, simplification helps to streamline problems and solutions, allowing us to solve equations and perform calculations more efficiently.