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Solving the Exponential Equation: (0.5)^2x = 2^(1-3x)
This article will guide you through the steps of solving the exponential equation: (0.5)^2x = 2^(1-3x).
Understanding the Equation
The equation involves exponential expressions with different bases. To solve it, we need to manipulate the equation to have the same base on both sides.
Solution Steps
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Express both sides with the same base:
- We can rewrite 0.5 as 1/2.
- Since 2 is the base of both sides, we can express (1/2) as 2^(-1).
Therefore, the equation becomes: (2^(-1))^2x = 2^(1-3x)
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Simplify using exponent rules:
- When raising a power to another power, we multiply the exponents.
This gives us: 2^(-2x) = 2^(1-3x)
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Equate the exponents:
- Now that we have the same base on both sides, we can equate the exponents.
This leads to: -2x = 1 - 3x
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Solve for x:
- Add 3x to both sides: x = 1
Conclusion
The solution to the exponential equation (0.5)^2x = 2^(1-3x) is x = 1.
Remember, it's crucial to understand the rules of exponents to effectively solve these types of equations. Practice with various examples to enhance your understanding and problem-solving skills.