Problem 32 - Algebra A Combined Function

Question

Use the product rule to simplify each expression. $$ \left(12 x^{5}\right)\left(-x^{6}\right)\left(x^{4}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-12x^{15}\).

Step 1: Identify the Expression to Simplify

The expression given is \((12 x^{5})(-x^{6})(x^{4})\). We need to use the product rule of exponents to simplify this expression.

Step 2: Rearrange the Factors

Reorganize the expression to group the coefficients and the powers of \(x\) separately: \(12 \cdot (-1) \cdot (x^{5} \cdot x^{6} \cdot x^{4})\).

Step 3: Multiply the Coefficients

Calculate the product of the numerical coefficients: \(12 \times (-1) = -12\).

Step 4: Apply the Product Rule for Exponents

Use the product rule for exponents, which states that \(x^a \cdot x^b = x^{a+b}\). Apply it to \(x^{5} \cdot x^{6} \cdot x^{4}\): \(x^{5+6+4} = x^{15}\).

Step 5: Combine the Results

Multiply the result of the coefficients with the simplified exponent term: the simplified expression becomes \(-12x^{15}\).

Key Concepts

Algebraic Expressions

When dealing with algebraic expressions, we often encounter terms composed of numbers and variables. An algebraic expression can include numbers, variables like \(x\), and mathematical operations (like addition, multiplication, etc.). These expressions allow us to represent quantities in a flexible way and are fundamental in solving various algebraic problems.
One key aspect of working with algebraic expressions is understanding how each component works together. Let's break it down:

• **Coefficients**: These are the numerical parts of the terms. In our example, \(12\) and \(-1\) are coefficients.
• **Variables**: Symbols used to represent unknown values. In our expression, \(x\) is the variable.
• **Exponents**: Dictate how many times a variable is multiplied by itself. For instance, \(x^5\) means \(x\) is multiplied by itself five times.

Understanding these components helps us simplify expressions and solve equations more effectively.

Simplifying Expressions

Simplifying expressions is the process of making them as compact and efficient as possible. This often involves combining like terms and applying mathematical rules to reduce the expression to its simplest form.
In our exercise, the goal is to use the product rule to simplify the expression \((12 x^{5})(-x^{6})(x^{4})\). Here's what simplification involves in this context:

• **Rearranging and Grouping Similar Terms**: We first put coefficients together and group like bases of variables. Here: \(12 \cdot (-1)\) gives us the coefficient product, while \(x\) terms are grouped together.
• **Using Rules of Exponents**: By applying the product rule for exponents, we combine the powers of \(x\), hence \(x^{5 + 6 + 4}\) becomes \(x^{15}\).

Simplification makes expressions easier to work with and solves complex problems efficiently by reducing the chances of errors.

Exponents

Exponents are a crucial part of algebra that allows us to concisely represent repeated multiplication of the same number by itself. In expressions like \(x^{5}\), the 5 is an exponent indicating how many times \(x\) is used as a factor.
When simplifying expressions with exponents, such as in our exercise, the **Product Rule for Exponents** is key:

• **Product Rule**: If you multiply powers of the same base, you can add the exponents: \(x^a \cdot x^b = x^{a+b}\). This rule helps simplify expressions like \(x^5 \cdot x^6 \cdot x^4\) to \(x^{15}\).
• **Negative Coefficients**: While exponents apply to variables, remember to handle numeric parts separately; e.g., \(12 \times (-1)\) must be computed independently.

Understanding how to apply the product rule and control coefficients appropriately simplifies calculations and clarifies many algebraic tasks. Exponents transform complex multiplicative operations into short, manageable forms, essential for solving advanced math problems efficiently.