Problem 50 - A Graphical Approach to College Algebra

Question

Solve each equation. $$\left(\frac{1}{2}\right)^{3 x-6}=8^{x+1}$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{2}\).

Step 1: Rewrite both sides with the same base

To solve the equation \(\left(\frac{1}{2}\right)^{3x-6}=8^{x+1}\), we need to express both sides with the same base. Notice that \(\frac{1}{2}\) can be written as \(2^{-1}\) and \(8\) can be written as \(2^3\). So, rewrite the equation: \[(2^{-1})^{3x-6} = (2^3)^{x+1}\]

Step 2: Apply the power of a power property

Using the power of a power property \((a^m)^n = a^{mn}\), simplify both sides of the equation:The left side becomes \(2^{-1(3x-6)} = 2^{-3x+6}\).The right side becomes \(2^{3(x + 1)} = 2^{3x + 3}\).Thus, the equation becomes: \[2^{-3x+6} = 2^{3x+3}\]

Step 3: Set the exponents equal

Since the bases are the same, we can set the exponents equal to each other:\[-3x + 6 = 3x + 3\]

Step 4: Solve for \(x\)

To find \(x\), solve the equation:Add \(3x\) to both sides: \[6 = 6x + 3\] Subtract \(3\) from both sides:\[3 = 6x\]Finally, divide both sides by \(6\):\[x = \frac{1}{2}\]

Key Concepts

Power of a Power Property

When you encounter expressions with exponents raised to another power, the 'Power of a Power Property' is what you need! This property is useful because it allows you to simplify complicated exponent expressions. In mathematical terms, if you have \((a^m)^n\), you can simplify it to \(a^{mn}\).
Let's break this down:

• "a" represents the base number.
• "m" and "n" are the exponents. They show how many times we multiply the base by itself.

This property provides a straightforward way to handle multiple exponents.
For instance, if you are computing \((2^{-1})^{3x - 6}\), it leads to \(2^{-1 \times (3x - 6)} = 2^{-3x + 6}\).
By simplifying complex power operations, this property makes your calculations neater and easier to manage.

Same Base Property

The 'Same Base Property' comes in handy when you have an equation where both sides have exponents with the same base. This property states that if \(a^m = a^n\), then the exponents \(m\) and \(n\) must be equal.
Simply put:

• If two identical base numbers are equal in terms of their powers, the powers themselves must be equal.

In our exercise, both sides of the equation are eventually expressed with base 2: \(2^{-3x + 6} = 2^{3x + 3}\).
This means we can equate the exponents directly: \(-3x + 6 = 3x + 3\).
By using the 'Same Base Property', the solution simplifies to solving just a linear equation for \(x\). It turns a problem that looks complex into something manageable.

Solving Equations

When you solve equations, especially those involving exponents, it is essential to handle them step by step. Once you've ensured that your equation uses the 'Power of a Power Property' and 'Same Base Property', the solving gets simplified.
Consider the equation \(-3x + 6 = 3x + 3\), which arises after applying these properties.
Here’s how you solve it:

• Add \(3x\) to both sides of the equation to gather all \(x\) terms on one side: \(6 = 6x + 3\).
• Next, subtract 3 from both sides to simplify: \(3 = 6x\).
• Finally, divide both sides by 6 to isolate \(x\), which gives \(x = \frac{1}{2}\).

This process demonstrates that solving equations is about simplifying and rearranging until you isolate the variable. By consistently applying these logical steps, you can solve complicated equations with confidence.