Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the expression. $$ (-2 x)^{3}(-2 x)^{2} $$

Step-by-Step Solution

Answer

The simplified expression for \((-2x)^3(-2x)^2\) is \((-2x)^5\).

Step 1: Identify the base and exponents

In this expression, the base is -2x. The exponents are 3 for the first factor and 2 for the second factor.

Step 2: Apply the product of powers property

The product of powers property states that \(a^{m}a^{n} = a^{m+n}\). Here, \(a=-2x\), \(m=3\), and \(n=2\). Therefore: \((-2x)^3(-2x)^2 = (-2x)^{3+2}\)

Step 3: Add the exponents

We need to add the exponents 3 and 2: \(3 + 2 = 5\)

Step 4: Write the simplified expression

Now we have the simplified expression: \((-2x)^3(-2x)^2 = (-2x)^5\)

Key Concepts

Product of Powers Property

When dealing with powers, understanding the "product of powers" property can be incredibly helpful. This mathematical rule allows you to simplify expressions with the same base. The rule is pretty straightforward: if you have two exponents with the same base, you simply add the exponents together. The general formula is \(a^m \times a^n = a^{m+n}\). Here's a simple breakdown:

• The base (often noted as \(a\)) stays the same throughout the equation.
• The exponents (\(m\) and \(n\)) are added together to form a new exponent.

For example, if you have \((-2x)^3 \cdot (-2x)^2\), since the base \((-2x)\) is the same, apply the property to get \((-2x)^{3+2} = (-2x)^5\). This property makes it easier to manage and simplify expressions rather than expanding each term.

Simplifying Expressions

Simplifying expressions is all about making them easier to work with. It involves taking a complex expression and rewriting it so that it's more comprehensible. This often means reducing the number of terms or combining like terms. You want to find a version that is both accurate and easy to use. Here are the steps you generally follow:

• Identify like terms or common bases within the expression.
• Apply relevant mathematical properties or rules, such as the product of powers.
• Combine the information into a more streamlined formula.

Think of it as cleaning up your work to make calculations simpler. For instance, in our original problem, the expression \((-2x)^3 \cdot (-2x)^2\) can be simplified using our product of powers property to \((-2x)^5\). This makes working with the expression far more manageable and helps avoid cumbersome calculations.

Mathematical Expressions

At the heart of math, mathematical expressions are a collection of numbers, variables, and operators that represent a value or a relationship between values. Expressions can range from simple numbers, like \(5+3\), to complex formulas involving multiple variables and exponents.Here's a quick guide:

• Numbers are constant values, like \(5\) or \(-2\).
• Variables are symbols that stand in for numbers and can change, like \(x\).
• Operators tell you what to do with the numbers or variables, like addition \(+\), subtraction \(-\), multiplication \(\times\), and division \(\div\).
• Exponents denote how many times a base is multiplied by itself.

In mathematical expressions like \((-2x)^3\times(-2x)^2\), you use a combination of operators and powers to reach a simplified version. Understanding each component helps break down more challenging problems into something manageable. Remember, the goal is to translate complicated math speak into something both functional and approachable. This assists in both solving the problem at hand and in grasping broader mathematical concepts.