Problem 4 - College Algebra with Corequisite Support

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 is \(v = -1\).

Step 1: Identify the Like Bases

In the given equation, both sides have the base 4. As the bases are the same, we can set the exponents equal to each other to solve for the variable.

Step 2: Set the Exponents Equal to Each Other

Since the bases are the same, we set the exponents equal to each other: \[-3v - 2 = -v\]

Step 3: Simplify the Exponent Equation

To isolate the variable \(v\), we start by adding \(3v\) to both sides of the equation:\[-3v - 2 + 3v = -v + 3v\]This simplifies to:\[-2 = 2v\]

Step 4: Solve for the Variable

Now, solve for \(v\) by dividing both sides by 2:\[\frac{-2}{2} = \frac{2v}{2}\]This simplifies to:\[v = -1\]

Step 5: Verify the Solution

Substitute \(v = -1\) back into the original exponents to ensure both sides are equal:Left side: \(4^{-3(-1)-2} = 4^{3-2} = 4^1 = 4\)Right side: \(4^{-(-1)} = 4^1 = 4\)Since both sides are equal, the solution \(v = -1\) is verified.

Key Concepts

Understanding Exponents

Exponents are a way of expressing repeated multiplication of the same number. They consist of a base and an exponent; the base is the number being multiplied, while the exponent indicates how many times to multiply that base by itself. For example:

• If the base is 4 and the exponent is 3, then the expression is written as \(4^3\) and means \(4 \times 4 \times 4\).

Exponents can also be negative, which means they represent reciprocal powers. For example, \(4^{-1}\) is equal to \(\frac{1}{4}\) because it's the same as \(\frac{1}{4^1}\). Learning to manipulate these forms is crucial for solving exponential equations.

Using Like Bases

When solving exponential equations, like bases can be powerful tools. If the bases on either side of the equation are the same, you can simplify the equation by setting the exponents equal to each other. In the problem we have, both sides of the equation use base 4:

• Left Side: \(4^{-3v-2}\)
• Right Side: \(4^{-v}\)

Since the bases are identical, you can set \(-3v - 2\) equal to \(-v\), making it easier to focus solely on manipulating the exponents to solve for the variable.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging an equation to isolate the variable. In our example, after setting the exponents equal, we need to simplify:

• Start by adding \(3v\) to both sides: \(-3v - 2 + 3v = -v + 3v\), which simplifies to \(-2 = 2v\).
• Now, solve for \(v\) by dividing both sides by 2: \(\frac{-2}{2} = \frac{2v}{2}\).

This gives the solution \(v = -1\). The key to algebraic manipulation is performing the same operation on both sides of the equation to maintain equality.

Verification of Solutions

After finding a solution, it’s crucial to verify that it satisfies the original equation. Substitute \(v = -1\) back into the exponents:

• Left Side: Substitute \(-1\) for \(v\) in \(-3v-2\) to get \(4^1 = 4\).
• Right Side: Substitute \(-1\) for \(v\) in \(-v\) to get \(4^1 = 4\).

Both sides equate to 4, confirming that \(v = -1\) is indeed the correct solution. Verification ensures consistency and accuracy in your results, forming a critical part of solving mathematical equations.