Problem 71 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ \frac{2(-4-2 \cdot 2)}{3(-3)(-2)} $$

Step-by-Step Solution

Answer

The expression evaluates to \(-\frac{8}{9}\).

Step 1: Evaluate the Expression Inside the Parentheses

Start with the numerator by simplifying the expression inside the parentheses. Calculate \(-4 - 2 \cdot 2\). 1. First, perform the multiplication: \(2 \cdot 2 = 4\).2. Next, substitute the result back into the expression: \(-4 - 4 = -8\). So, the expression inside the numerator becomes \(-8\).

Step 2: Multiply and Simplify the Numerator

Now that we have simplified the inside of the parentheses in the numerator, multiply it by 2: \[2 \times (-8) = -16\].

Step 3: Evaluate the Denominator

Simplify the denominator by performing the multiplication: 1. Multiply the first two terms: \(3 \times (-3) = -9\).2. Then, multiply the result by the third term: \(-9 \times (-2) = 18\).Thus, the denominator is simplified to \(18\).

Step 4: Divide the Simplified Numerator by the Denominator

Now, divide the numerator by the denominator:\[\frac{-16}{18}\].

Step 5: Simplify the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) for \(-16\) and \(18\), which is \(2\):\[\frac{-16 \div 2}{18 \div 2} = \frac{-8}{9}\].This makes \(-\frac{8}{9}\) the simplest form of the fraction.

Key Concepts

Numerical Expression Evaluation

Evaluating a numerical expression involves solving it to find its numerical value. This exercise requires evaluating the expression \( \frac{2(-4-2 \cdot 2)}{3(-3)(-2)} \), where various operations such as multiplication and addition are involved. It’s essential to work through these operations step by step to avoid errors.

Follow these steps for a successful evaluation:

• Identify and prioritize operations based on mathematical rules. Parentheses are addressed first, followed by multiplication and division.
• Start with the innermost expression, simplifying it to its simplest form - in this case, \(-4 - 2 \cdot 2\) is simplified to \(-4 - 4\), resulting in \(-8\).
• Apply multiplication next, turning it into \(2 \times (-8)\).

Approaching problems in an organized fashion can drastically simplify the evaluation of such expressions, enabling a smoother path to the solution.

Fractions Simplification

Simplifying fractions is a crucial aspect of working with algebraic expressions. It helps convert complex fractions into their simplest form, which is easier to interpret and work with. In our exercise, we encounter the fraction \( \frac{-16}{18} \). Breaking it down into simpler parts ensures better understanding.

To successfully simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator. For \(-16\) and \(18\), the GCD is \(2\).
• Divide both the numerator and the denominator by their GCD. This transforms \( \frac{-16}{18} \) to \( \frac{-8}{9} \).
• Simplifying is essential for clearer solutions and is a standard practice in algebra to present the cleanest answer possible.

By ensuring fractions are reduced to their simplest form, we not only achieve the exercise's goal but also adhere to mathematical clarity.

Order of Operations

The order of operations is a fundamental rule set in mathematics, guiding how to evaluate expressions correctly. This rule, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), ensures calculations are performed in the right sequence.

Applying it to our example expression:

• Begin with the evaluation within parentheses \(-4 - 2 \cdot 2\). This is prioritized over basic arithmetic outside.
• Afterward, progress to multiplication such as \(2 \times (-8)\) and subsequently \(3(-3)(-2)\), ensuring each multiplication follows the evaluated parentheses.
• This structured approach eliminates the potential errors induced by disorderly calculations, reinforcing the proper handling of every term.

Mastering and applying the order of operations is crucial for solving any algebraic expression accurately, thus preventing disputes over arithmetic results.