Problem 11 - Precalculus: Functions and Graphs

Question

Simplify. $$\left(\frac{1}{2} x^{4}\right)\left(16 x^{5}\right)$$

Step-by-Step Solution

Answer

\( 8x^9 \)

Step 1: Distribute Constant Multiplication

Multiply the constants outside the variables: \( \frac{1}{2} \) and \( 16 \). Calculate \( \frac{1}{2} \times 16 = 8 \).

Step 2: Apply Exponent Multiplication Rule

Use the rule \( x^a \times x^b = x^{a+b} \). Here, multiply \( x^4 \) and \( x^5 \) to get \( x^{4+5} = x^9 \).

Step 3: Combine Results

Combine the results from Steps 1 and 2. You get \( 8 \times x^9 \), or simply \( 8x^9 \).

Key Concepts

Constant Multiplication

In mathematics, constant multiplication involves multiplying numerical constants together before dealing with variable parts. In our exercise, we have the constants \( \frac{1}{2} \) and \( 16 \). These two numbers are multiplied first, independent of the variables. The process simplifies the expression by focusing only on set numbers.
By multiplying \( \frac{1}{2} \) and \( 16 \), you get \( 8 \). This results because \( \frac{1}{2} \) is essentially slicing something in half, so multiplying it by \( 16 \) is like taking half of \( 16 \), which is \( 8 \).

This approach reduces complexity in expressions involving variables and is an essential step in accurately and efficiently simplifying algebraic expressions.

Product Rule

The product rule for exponents is a powerful tool that helps simplify expressions involving powers of the same base. This rule states: \( x^a \times x^b = x^{a+b} \). It allows you to combine exponents when multiplying two powers with the same base.
In the given exercise, the bases are the same (both are \( x \)). Thus, apply the rule to \( x^4 \) and \( x^5 \). Combine them into one power, resulting in \( x^{4+5} \), which simplifies to \( x^9 \).

• Keep the base (\( x \)) the same.
• Add the exponents \( 4 + 5 \).

This rule is a principal concept in exponentiation, as it helps in consolidating similar terms, making the expression simpler and easier to understand.

Simplification

Simplification is the process to make an expression more concise while retaining its original value. After you've handled constant multiplication and the product rule, the final result of the given expression is \( 8x^9 \). Combining all earlier steps in a neat package reflects the solution's correctness.
Simplification involves:

• Combining constant multiplication results: \( 8 \).
• Applying the product rule: \( x^9 \).
• Merging these components: \( 8x^9 \).

Simplifying not only makes expressions easier to read and use, but it ensures that you're left with the most streamlined version of your mathematical work.