Problem 4 - College Algebra

Question

For the following exercises, use like bases to solve the exponential equation. $$ 4^{-3 v-2}=4^{-v} $$

Step-by-Step Solution

Answer

The solution to the equation is \(v = -1\).

Step 1: Identify Like Bases

In the given equation, both sides have the base 4: \(4^{-3v-2} = 4^{-v}\). Since the bases are the same, we can equate the exponents.

Step 2: Set the Exponents Equal

Equate the exponents of the powers: \(-3v - 2 = -v\). This is because \(a^b = a^c\) implies \(b = c\) if base \(a\) is non-zero.

Step 3: Simplify the Equation

Rearrange the equation to solve for \(v\). Start by adding \(3v\) to both sides to get:\(-2 = 2v\).

Step 4: Solve for v

Divide both sides by \(2\) to isolate \(v\): \(v = -1\).

Key Concepts

Like Bases

When working with exponential equations, recognizing like bases is crucial. A base is the number being raised to a power, or exponent, and it’s the same on both sides of some equations. In the original exercise, the base is 4 on both sides: \(4^{-3v-2} = 4^{-v}\).

• By having like bases, you can focus on the exponents, knowing that the exponential expressions are set equal due to their common base.
• This allows simplification by equating exponents directly.
• The concept of like bases simplifies solving significantly.

Once you’ve identified the like bases, the next step is to work with the exponents to find the solution. This approach avoids complicated operations and keeps the process straightforward.

Exponent Rules

Exponential equations are governed by rules that dictate how exponents interact. A critical rule for solving equations with like bases is that if two powers with the same base are equal, then their exponents must be equal. Mathematically, this is expressed as: if \(a^b = a^c\), then \(b = c\).

• This rule is applied when we equate the exponents in the given exercise: \(-3v-2 = -v\).
• Exponents can be manipulated using common algebraic techniques, such as adding, subtracting, multiplying, or dividing.
• Understanding these operations helps in solving complex equations efficiently.

Therefore, mastering exponent rules streamlines the process from equating exponents to solving the equation. This enables a deeper dive into solutions once the exponents have been isolated.

Equating Exponents

Once you identify that the bases are like, the next step in solving exponential equations is equating the exponents. In this scenario, it means treating the exponents as a standard algebraic equation: \(-3v-2 = -v\).

• By focusing on the exponents, you transform an exponential problem into a simpler algebraic one.
• Adjust the equation by isolating the variable term, in this case by adding \(3v\) to both sides.
• The manipulation leads to solving for the variable, which in the exercise simplifies to \(v = -1\).

Equating exponents turns the task into a more familiar process, leveraging basic algebra to unravel the unknown variable. This strategy is efficient and reduces potential errors by focusing on core arithmetic operations.