Problem 81 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ \frac{|6-4|+2|-4|}{226-6^{3}} $$

Step-by-Step Solution

Answer

The value of the expression is 1.

Step 1: Calculate the Absolute Values

First, evaluate the absolute values in the numerator. The expression is \(|6-4|+2|-4|\):- \(|6-4| = |2| = 2\)- \(|-4| = 4\)So the numerator becomes: \(2 + 2 \times 4\).

Step 2: Simplify the Numerator

Let's simplify the numeral expression:- First, perform the multiplication: \(2 \times 4 = 8\)So, \(2 + 8 = 10\).Now, the numerator is \(10\).

Step 3: Simplify the Denominator

Next, evaluate the expression in the denominator: \(226 - 6^3\).First, calculate \(6^3 = 6 \times 6 \times 6 = 216\).So, the denominator becomes \(226 - 216 = 10\).

Step 4: Divide the Numerator by the Denominator

Now divide the simplified numerator by the simplified denominator:\(\frac{10}{10} = 1\).

Key Concepts

Absolute Value

Absolute value is a concept in mathematics that measures the distance of a number from zero on the number line. It is always a non-negative value because distance cannot be negative. To solve the exercise, we first find the absolute values of given numbers within the expression.

• For the expression \(|6-4|\), we begin by subtracting 4 from 6, yielding 2. Taking the absolute value, we get \(|2| = 2\).
• For \(|-4|\), since distance is positive, we take 4 as the absolute value, so \(|-4| = 4\).

These steps are crucial in simplifying expressions that include absolute values as they ensure we consider only their magnitudes, discarding any sign the numbers might originally have.

Numerators and Denominators

Numerators and denominators play a vital role when dealing with fractions. The numerator is the top part, representing the number of parts we have or need to consider. The denominator is the bottom part, representing the total number of equal parts.
In the provided expression \(\frac{|6-4|+2|-4|}{226-6^3}\), once the absolute values are calculated, we simplify the numerator:

• Starting from \(2 + 2 \times 4\), you perform multiplication first, following the order of operations, giving us 8.
• Add the result to 2, resulting in a simplified numerator of 10.

Meanwhile, for the denominator, we evaluate closely the expression \(226-6^3\). Calculating \(6^3 = 216\), we then subtract this from 226 to simplify it to 10. Understanding how to manage numerators and denominators directly impacts the simplification of the overall expression.

Order of Operations

The order of operations is a fundamental principle in mathematics, guiding in which sequence to evaluate parts of an expression. Often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
In our expression, both the numerator and denominator needed careful evaluation guided by this principle:

• Within the numerator, \(|6 - 4| + 2 \times |-4|\), compute the absolute values first, then execute multiplication before addition, complying with the order of operations.
• In the denominator, \(226 - 6^3\), after evaluating exponents (where \(6^3 = 216\)), subtraction simplifies the expression further.

Applying these rules systematically ensures that expressions are evaluated correctly, leading to the right final answer.