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Magazine Chapter 12: Problem 34
Solve. $$ 4^{3 x-7}=32^{2 x} $$
Short Answer
Expert verified
\(x = -\frac{7}{2}\)
Step by step solution
01
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: \[ (2^2)^{3x-7} = (2^5)^{2x} \]
02
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: \[ 2^{6x - 14} = 2^{10x} \]
03
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: \[ 6x - 14 = 10x \]
04
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: \[ x = -\frac{7}{2} \]
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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: \((2^5)^{2x} = 2^{5 \times 2x} = 2^{10x}\)
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:
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 \(x = -\frac{7}{2}\).
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 \(32^{2x}\).
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 \((2^5)^{2x}\).
