Problem 19 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(\frac{(4 x)^{3}}{4 x^{3}}\)

Step-by-Step Solution

Answer

The expression simplifies to 16.

Step 1: Understanding the Problem

We are given the expression \( \frac{(4x)^3}{4x^3} \) to simplify, where all variables are considered positive. This involves simplifying both the numerator and the denominator separately.

Step 2: Expanding the Numerator

The numerator of the expression is \((4x)^3\). To expand this, use the property of exponents that states \((ab)^n = a^n b^n\). We have \((4x)^3 = 4^3 \cdot x^3 = 64x^3\).

Step 3: Simplifying the Fraction

Now the expression becomes \( \frac{64x^3}{4x^3} \). We can simplify this fraction by cancelling out common terms from the numerator and the denominator. The terms \(x^3\) in both the numerator and the denominator cancel each other out.

Step 4: Simplifying the Remaining Expression

With \(x^3\) cancelled, we are left with \( \frac{64}{4} \). Divide 64 by 4 to simplify the expression: \( \frac{64}{4} = 16 \). Thus, the expression simplifies to 16.

Key Concepts

Understanding Exponent Rules

Exponents can initially seem tricky, but following a few fundamental rules makes them much easier to handle. When simplifying expressions with exponents, you often encounter the 'Power of a Product' rule. This rule tells us that when you have \((ab)^n = a^n imes b^n\). It's a straightforward principle that states you apply the exponent to each factor inside the parentheses. For instance, let's look at the numerator in our exercise: \((4x)^3\).Using the rule, this expands to:

• \(4^3 imes x^3 = 64x^3\)

Key to applying exponent rules correctly is practicing such expansions using these simplify guiding principles. Ensure to double-check each term's expansion, especially when multiple factors are involved.Remember, the rules remain consistent regardless of the numbers or variables involved, simplifying the application process.

Fraction Simplification Techniques

Simplifying fractions is an essential skill when tackling algebraic expressions. The method is to cancel common factors in the numerator and the denominator. Let’s work through the example \( \frac{64x^3}{4x^3} \).1. **Identify Common Factors:** The terms \(x^3\) appear in both the numerator and denominator, allowing them to cancel out. After this cancellation, you get: \( \frac{64}{4} \).2. **Simplify Remaining Values:** Simplifying \( \frac{64}{4} \) gives you 16.

• This step involves basic division. Divide the top number by the bottom number.
• Ensure there are no more common factors left to cancel before concluding simplification.

By following these steps meticulously, you eliminate complexities within any fractional form. The process turns potentially daunting fractions into much more manageable figures.

Role of Positive Variables

In algebraic simplification, it’s critical to note when variables are restricted to positive values. Given the premise that variables are positive in the exercise, it streamlines the simplification:

• No need to worry about negative exponents or zero-denominator scenarios, which can complicate the process.

Moreover, ensuring variables are positive helps when cancelling terms. Any incorrect cancellations (like dividing by zero) aren’t a concern here:

• For example, knowing \(x^3\) is positive ensures direct and valid cancellations occurred in the previous section.

Positive constraints simplify checking the legitimacy of results. They assure that each step in calculations remains on the proper path without unintended complexity from negative numbers.