Problem 70 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. $$ 7(2)(-w)(-w) \text { when } w=6 $$

Step-by-Step Solution

Answer

The value of the expression when \(w = 6\) is \(504\).

Step 1: Substituting the value of variable

In the given expression, replace the variable \(w\) with its given value, which is 6. So, the expression becomes \(7(2)(-6)(-6)\).

Step 2: Multiply the numbers and solve the expression

First, multiply \(2\) and \(7\) to get \(14\). So, the expression becomes \(14(-6)(-6)\). Then, multiply \(14\) by \(-6\) twice, because \(-6\) occurs twice, which results in \(504\).

Step 3: Final Answer

Evaluating the expression, we find it equals 504.

Key Concepts

Substitution

Substitution in algebra involves replacing a variable in an expression with a specific value. This is a fundamental concept often used in solving algebraic problems. Substitution allows you to simplify an expression or equation by making it concrete. In our exercise, the variable is \(w\), and we are given \(w=6\). So, wherever \(w\) appears in the expression, it is replaced with 6. The original expression is \(7(2)(-w)(-w)\), and after substitution, it becomes \(7(2)(-6)(-6)\). Substitution is a straightforward process, but it's vital to ensure that each instance of the variable is replaced to maintain the integrity of the expression. Once substitution is complete, you have a numerical expression ready for simplification through other operations.

Multiplication

Multiplication is one of the basic operations in mathematics and is essential in many algebraic solutions. It involves combining equal groups together. In expressions, multiplication is indicated by placing numbers or variables side by side, usually within parentheses. In our expression, we first calculate \(7 \cdot 2\), combining the numbers from left to right. The result is 14. The expression hence becomes \(14(-6)(-6)\). Next, we multiply 14 by \(-6\), which we encounter twice. It's crucial to proceed step-by-step, multiplying each number in sequence to keep track of the results. This ensures accuracy and helps in understanding how the expression changes at each step.

Integer Operations

Integer operations encompass addition, subtraction, multiplication, and division of whole numbers, which include negative numbers. In this exercise, multiplication is emphasized, especially involving negative integers. When multiplying integers:

• Multiplying two positive integers gives a positive result.
• Multiplying a positive integer by a negative integer gives a negative result.
• Multiplying two negative integers gives a positive result.

In the expression \(14(-6)(-6)\), first multiply 14 by \(-6\), resulting in \(-84\). Then, multiply \(-84\) by \(-6\) again. The rule states that two negatives make a positive, leading to the final positive result of 504. Understanding the rules of integer multiplication is crucial for getting the correct results and helps eliminate confusion with signs.