Solve: [tex]9^6=\left(3^2\right)^{2 x-2}[/tex]
Answer (2)
Rewrite the equation using the property 9 = 3 2 , resulting in ( 3 2 ) 6 = ( 3 2 ) 2 x − 2 .
Simplify both sides of the equation using the power of a power rule, obtaining 3 12 = 3 4 x − 4 .
Equate the exponents since the bases are equal: 12 = 4 x − 4 .
Solve for x : x = 4 16 = 4 . The final answer is 4 .
Explanation
Understanding the Problem We are given the equation 9 6 = ( 3 2 ) 2 x − 2 and asked to solve for x . We will use properties of exponents to simplify the equation and isolate x .
Rewriting the Equation First, rewrite the equation using the property 9 = 3 2 . This gives us ( 3 2 ) 6 = ( 3 2 ) 2 x − 2 .
Simplifying the Left Side Next, simplify the left side of the equation using the power of a power rule, ( a m ) n = a mn . So, ( 3 2 ) 6 = 3 2 × 6 = 3 12 .
Simplifying the Right Side Similarly, simplify the right side of the equation: ( 3 2 ) 2 x − 2 = 3 2 ( 2 x − 2 ) = 3 4 x − 4 .
Equating the Exponents Now the equation is 3 12 = 3 4 x − 4 . Since the bases are equal, the exponents must be equal. Therefore, 12 = 4 x − 4 .
Solving for x Solve for x : Add 4 to both sides of the equation to get 12 + 4 = 4 x , which simplifies to 16 = 4 x . Divide both sides by 4 to isolate x : x = 4 16 = 4 .
Verification Finally, we can check our solution by substituting x = 4 into the original equation: 9 6 = ( 3 2 ) 2 ( 4 ) − 2 = ( 3 2 ) 8 − 2 = ( 3 2 ) 6 = 9 6 . The solution is valid.
Final Answer Therefore, the solution to the equation 9 6 = ( 3 2 ) 2 x − 2 is x = 4 .
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, they can model population growth, radioactive decay, and compound interest. Understanding how to solve exponential equations allows us to predict future values, determine decay rates, and calculate investment returns. In finance, compound interest is calculated using an exponential formula, where the amount of money accumulated grows exponentially over time. By solving exponential equations, we can determine the time it takes for an investment to reach a certain value or calculate the interest rate needed to achieve a financial goal.
The solution to the equation 9 6 = ( 3 2 ) 2 x − 2 is x = 4 . We simplified the equation by expressing 9 as 3 2 and equated the exponents. The verification step confirmed the accuracy of the solution.
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