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(iii) 4^(x-1)times(0.5)^(3-2x)=((1)/(8))^(x)

3 min read Jun 16, 2024
(iii) 4^(x-1)times(0.5)^(3-2x)=((1)/(8))^(x)

Solving the Exponential Equation: 4^(x-1) * (0.5)^(3-2x) = (1/8)^x

This article will guide you through solving the exponential equation: 4^(x-1) * (0.5)^(3-2x) = (1/8)^x. We will utilize the properties of exponents to simplify the equation and ultimately find the solution for x.

Understanding the Properties of Exponents

Before we dive into the solution, let's recall some key properties of exponents that will be crucial in simplifying the equation:

  • Product of powers: a^m * a^n = a^(m+n)
  • Quotient of powers: a^m / a^n = a^(m-n)
  • Power of a power: (a^m)^n = a^(m*n)

Simplifying the Equation

  1. Express all terms with the same base:

    • Notice that 4, 0.5, and 1/8 can all be expressed as powers of 2:

      • 4 = 2^2
      • 0.5 = 1/2 = 2^(-1)
      • 1/8 = 2^(-3)

    • Substitute these values into the original equation:

      • (2^2)^(x-1) * (2^(-1))^(3-2x) = (2^(-3))^x

  2. Apply the power of a power rule:

    • (2^(2*(x-1))) * (2^(-1*(3-2x))) = 2^(-3x)
    • 2^(2x-2) * 2^(-3+2x) = 2^(-3x)

  3. Apply the product of powers rule:

    • 2^(2x-2 -3 + 2x) = 2^(-3x)
    • 2^(4x - 5) = 2^(-3x)

  4. Equate the exponents:

    • Since the bases are now the same, we can equate the exponents:
      • 4x - 5 = -3x

  5. Solve for x:

    • 7x = 5
    • x = 5/7

Conclusion

Therefore, the solution to the equation 4^(x-1) * (0.5)^(3-2x) = (1/8)^x is x = 5/7.

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