Problem 48 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. (Lessons 1.2,1.3) $$ 2\left(3^{3}-20\right) $$

Step-by-Step Solution

Answer

The evaluation of the expression \(2 \times (3^3 - 20)\) gives the result 14.

Step 1: Evaluate the expression inside the brackets

Start by evaluating \(3^3\), which results in \(27\). So, the expression inside the brackets becomes \(27 - 20\). Now perform this subtraction: \(27 - 20 = 7\). Thus, the expression reduces to: \(2 \times 7\).

Step 2: Multiply the result by 2

After evaluating the expression inside the brackets, multiply the result by 2: \(2 \times 7 = 14\).

Key Concepts

Order of Operations

When evaluating expressions like \(2(3^{3}-20)\), it's crucial to follow the proper sequence, known as the "Order of Operations." This helps to avoid confusion and errors. The standard order is:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This sequence can be easily remembered with the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Starting with operations inside brackets or parentheses simplifies expressions step by step. Here, we first computed \(3^3\) within the brackets before any other operations. This ensures precision in calculations.

Exponents

Exponents represent repeated multiplication of a base number. For instance, in \(3^3\), the exponent is 3. This means you multiply the base 3 by itself three times:

• \(3 imes 3 imes 3 = 27\)

Exponents are handled right after parentheses in order of operations. Once resolved, they simplify calculations significantly
and make it easier to manage expressions involving large numbers. Within our initial expression, \(3^3\) simplifies to 27, reducing complexity of the problem.

Arithmetic Operations

The fundamental arithmetic operations involved in this exercise are multiplication and subtraction, both integral to evaluating expressions. Here's a closer look:

1. **Subtraction**: Found inside our original expression in \(27 - 20\). It simplifies what's inside the brackets to 7 after resolving the exponent.
2. **Multiplication**: After simplifying the brackets, the operation outside is \(2 \times 7\), which finalizes the expression. This multiplication brings the result to 14, completing the computation.

Keeping in mind the order of operations ensures these steps are executed correctly, verifying accuracy in solving mathematical expressions.