%5Ex%3D64%2F27.jpeg "(3/4)^x=64/27")
Solving the Equation (3/4)^x = 64/27
This article will guide you through solving the equation (3/4)^x = 64/27. We'll break down the steps and provide explanations to help you understand the process.
Understanding the Equation
The equation (3/4)^x = 64/27 is an exponential equation. This means that the unknown variable, 'x', is in the exponent. To solve for 'x', we need to manipulate the equation to isolate 'x'.
Solving for 'x'
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Express both sides with the same base:
- Notice that both 64 and 27 are perfect cubes. We can rewrite them as 64 = 4³ and 27 = 3³.
- Therefore, the equation becomes (3/4)^x = (4/3)³.
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Simplify the equation:
- Since (4/3) is the reciprocal of (3/4), we can rewrite the equation as: (3/4)^x = [(3/4)⁻¹]³.
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Apply exponent rules:
- Using the rule (a^m)^n = a^(m*n), we simplify the right side: (3/4)^x = (3/4)⁻³.
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Equate exponents:
- Now that both sides have the same base, we can equate the exponents: x = -3.
Solution
Therefore, the solution to the equation (3/4)^x = 64/27 is x = -3.
Verification
To verify our solution, let's substitute x = -3 back into the original equation:
(3/4)^(-3) = (4/3)³ = 64/27
As you can see, the equation holds true when x = -3. This confirms our solution.
Conclusion
By understanding the properties of exponents and manipulating the equation strategically, we successfully solved the exponential equation (3/4)^x = 64/27. This process can be applied to solve other similar equations involving exponents.