Problem 34 - Algebra A Combined Function

Question

Solve. $$ 4^{3 x-7}=32^{2 x} $$

Step-by-Step Solution

Answer

\(x = -\frac{7}{2}\)

Step 1: Express Both Sides with the Same Base

The first step is to express both sides of the equation with the same base. Notice that both 32 and 4 can be expressed as powers of 2. We know that 4 is equal to \(2^2\) and 32 is equal to \(2^5\). Thus, rewrite the equation using these powers: \[ (2^2)^{3x-7} = (2^5)^{2x} \]

Step 2: Use the Power of a Power Property

Apply the power of a power property \((a^m)^n = a^{m imes n}\) on both sides of the equation: \[ 2^{2(3x-7)} = 2^{5 imes 2x} \]This simplifies to: \[ 2^{6x - 14} = 2^{10x} \]

Step 3: Set the Exponents Equal to Each Other

Since the bases on both sides are the same, the exponents must also be equal. This gives us the equation: \[ 6x - 14 = 10x \]

Step 4: Solve for x

Rearrange the equation to solve for \(x\):1. Subtract \(6x\) from both sides: \[ -14 = 4x \]2. Divide both sides by 4: \[ x = -\frac{14}{4} \]Simplify the fraction: \[ x = -\frac{7}{2} \]

Step 5: Verify the Solution

Substitute \(x = -\frac{7}{2}\) back into the original equation to verify:1. Compute the left side: \(4^{3(-\frac{7}{2})-7} = 4^{-\frac{21}{2}-7} = 4^{-\frac{35}{2}}\)2. Compute the right side: \(32^{2(-\frac{7}{2})} = 32^{-7} = 4^{-\frac{35}{2}}\)Since both sides are equal, the solution is verified.

Key Concepts

Power of a Power Property

When you encounter an expression like \((a^m)^n\), you can simplify it using the power of a power property.This property states that you multiply the exponents together: \((a^m)^n = a^{m \times n}\).
Let's see how this works in practice using part of our problem. Initially, we have:
\((2^2)^{3x-7}\) on one side and \((2^5)^{2x}\) on the other.
By applying the power of a power property, each side becomes:

• Left Side: \((2^2)^{3x-7} = 2^{2 \times (3x-7)} = 2^{6x-14}\)


• Right Side: \((2^5)^{2x} = 2^{5 \times 2x} = 2^{10x}\)

This makes equations much easier to manage because now the bases match, which leads directly to solving the exponents.

Exponent Rules

Exponents are powerful tools in mathematics, and understanding their rules is crucial for solving equations. The core rules used here involve simplifying and equating expressions with matching bases. In general, if you have an expression with matching bases, you can set the exponents equal to each other. This is because the only way the powers on each side stay equal is if their exponents are also equal.
For example, from our problem, after using the power of a power property, we end up with:\[2^{6x-14} = 2^{10x}\]
Since both sides of the equation have the same base \(2\), the exponents must be equal:

• Equate the exponents: \[6x - 14 = 10x\]


• Solve for \(x\) by rearranging: subtract \(6x\) from both sides, giving \(-14 = 4x\).
• Finally, divide by 4 to get \(x = -\frac{7}{2}\).

These rules streamline many seemingly complex problems into manageable steps.

Base Conversion

To solve certain exponential equations, it is often necessary to express both sides with the same base. Base conversion is a vital tactic here. Equations like the one in our problem become much simpler once each side of the equation shares a common base.
Let's illustrate the base conversion with our problem:
You have \(4^{3x-7}\) and \(32^{2x}\).

• Notice that both \(4\) and \(32\) are powers of \(2\).
• We express \(4\) as \(2^2\) and \(32\) as \(2^5\).
• This gives us: \((2^2)^{3x-7}\) and \((2^5)^{2x}\).

Converting bases ensures that we can simplify the equation by comparing the exponents directly. Once they are expressed as powers of the same base, using exponent rules makes solving for the variable a breeze. Base conversion is a foundational tool for solving complex exponential equations effectively.