(0 5)^2x=2^1-3x

2 min read Jun 16, 2024
(0 5)^2x=2^1-3x

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

  1. 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)

  2. Simplify using exponent rules:

    • When raising a power to another power, we multiply the exponents.

    This gives us: 2^(-2x) = 2^(1-3x)

  3. Equate the exponents:

    • Now that we have the same base on both sides, we can equate the exponents.

    This leads to: -2x = 1 - 3x

  4. 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.

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