Problem 5 - Elementary and Intermediate Algebra

Question

Fill in the blanks. If \(6^{4 x}=6^{-2},\) then \(4 x= \square\)

Step-by-Step Solution

Answer

The value of \(4x\) is \(-2\).

Step 1: Understand the Equation

We start with the equation \(6^{4x} = 6^{-2}\). We know that if the bases are the same, we can simply equate the exponents.

Step 2: Equate the Exponents

Since the bases are the same (both are 6), we equate the exponents: \(4x = -2\).

Step 3: Solve for x

We need to solve the equation \(4x = -2\) by isolating \(x\). Divide both sides by 4 to get \(x = \frac{-2}{4}\).

Step 4: Simplify the Result

Simplify \(\frac{-2}{4}\) to get \(x = -\frac{1}{2}\). Therefore, \(4x = -2\).

Key Concepts

Equating Exponents

When dealing with exponentiation, a key strategy is to equate exponents, particularly when the bases of the exponential terms are the same. This principle stems from the property that if two exponential expressions have identical bases, their exponents can be directly equated. This property simplifies algebraic equations significantly. In the original exercise, we have an equation with the same base: \(6^{4x} = 6^{-2}\). Given the matching bases of 6, the next logical step is to focus on the exponents.

By equating the exponents, we reduce the problem to something more manageable: \(4x = -2\). This step is crucial because it turns a potentially complex exponential equation into a simpler linear equation. Understanding this concept is pivotal for efficiently solving equations that involve powers and exponents.

Algebraic Equations

Algebraic equations are statements of equality involving variables and constants, expressed with mathematical operations. In this context, they form the backbone of solving problems related to exponentiation. The goal is to find the value of the variable that makes the equation true.

For the original exercise at hand, the exponential equation \(6^{4x}=6^{-2}\) simplifies to an algebraic equation \(4x = -2\) after equating the exponents. This shift allows us to apply basic algebraic techniques to uncover the solution.

• First, identify the terms with variables on both sides of the equation.
• Next, perform operations to isolate the variable of interest.
• Lastly, check the solution by substituting it back into the equation to ensure its validity.

This systematic approach is crucial in tackling various algebraic challenges, especially when they originate from exponentiation problems.

Equation Solving Steps

Solving equations, particularly those involving exponents, relies on a structured sequence of steps to ensure accuracy and simplify problem-solving. Understanding these steps builds a foundation for tackling more complex algebraic challenges. The original exercise demonstrated this with a clear process:

1. **Identify and Simplify:** Recognize that the exponents can be equated since the bases are the same. Simplify the equation to \(4x = -2\).

• Matching bases allow the problem to transform from an exponential to a linear format.

2. **Isolate the Variable:** The task now is to solve for \(x\).

• Divide both sides of the equation by 4, yielding \(x = \frac{-2}{4}\).

3. **Simplify the Solution:** Further simplify the fraction \(\frac{-2}{4}\) to \(x = -\frac{1}{2}\).

• Simplifying fractions is essential to getting the neatly reduced form of the solution.

This structured approach not only helps in solving the current problem efficiently but also equips learners with a template for future algebraic challenges involving exponents.