Problem 24 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. $$-2(8-4)$$

Step-by-Step Solution

Answer

The equivalent expression is \(-2(8) + (-2)(-4)\), which evaluates to \(-8\).

Step 1: Apply the Distributive Property

The Distributive Property states that for any numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). In this problem, we have \(-2\) as \(a\) and \((8-4)\) as \(b-c\). So we'll distribute \(-2\) to both terms inside the parenthesis. This means: \(-2\cdot8 + (-2)(-4)\).

Step 2: Perform the Multiplication

Now, we calculate each of the products obtained from the distribution. First, multiply \(-2\) by \(8\) to get \(-16\). Then, multiply \(-2\) by \(-4\), which gives \(+8\), since the product of two negative numbers is positive.

Step 3: Combine the Results

Add the two results from the multiplication: \(-16 + 8\).

Step 4: Evaluate the Expression

Finally, perform the addition: \(-16 + 8 = -8\).

Key Concepts

Equivalent Expressions

An equivalent expression is an expression that has the same value as another expression, even though they are written differently. Understanding how to create equivalent expressions is a fundamental part of algebra and can simplify problems or make calculations easier.

• One of the key methods to find equivalent expressions is using the Distributive Property, which allows us to simplify expressions by removing parentheses and rearranging terms.
• For instance, when working with \(-2(8-4)\), applying the Distributive Property gives us \(-2 \cdot 8 + (-2) \cdot (-4)\), transforming the initial expression into an equivalent but different-looking one.
• Equivalent expressions help verify the initial equation's correctness by giving the same value when evaluated, and they are crucial for solving equations.

Multiplication

Multiplication is one of the four fundamental operations in arithmetic and forms the basis for many algebraic concepts. In the context of algebra, multiplication can involve numbers, variables, or a combination of both.

• When distributing a term across an expression, multiplication comes into play by scaling each term inside the parentheses. For example, in the expression \(-2(8-4)\), we multiply each term in \(8-4\) by \(-2\).
• This process results in \(-2 \cdot 8\) and \(-2 \cdot (-4)\). It's important to perform each multiplication separately to ensure accuracy in the expression.
• Multiplication is commutative and associative, meaning the order in which numbers are multiplied doesn't affect the product: \(a \cdot b = b \cdot a\) and \((a \cdot b) \cdot c = a \cdot (b \cdot c)\). These properties can sometimes make calculations more straightforward.

Negative Numbers

Negative numbers are essential in mathematics and understanding their properties can help solve a wide range of problems.

• A negative number is usually indicated by a minus sign in front of it and represents a value less than zero. Operations involving negative numbers, such as multiplication or addition, follow specific rules.
• When multiplying two negative numbers, the result is positive. This is evident in our example where \(-2 \cdot (-4) = +8\). Understanding this result is crucial, as it affects the final outcome when solving equations.
• However, when multiplying a negative number by a positive number, the result is negative, like in \(-2 \cdot 8 = -16\).
• Knowing these rules helps in accurately computing expressions and in translating real-world problems into mathematical language.