Solve: $\left(\frac{1}{8}\right)^{-J a}=512^{3 a}$
Answer (2)
The solution to the equation ( 8 1 ) − J a = 51 2 3 a is found by rewriting the bases and equating the exponents. If J = 9 , the only solution is a = 0 . If J = 9 , any value of a is valid, but the explicit solution we can state is a = 0 .
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Rewrite the equation with a common base: ( 8 1 ) − J a = 51 2 3 a becomes 2 3 J a = 2 27 a .
Equate the exponents: 3 J a = 27 a .
Factor out a : a ( 3 J − 27 ) = 0 .
Solve for a : If J = 9 , then a = 0 . If J = 9 , then any value of a is a solution. Thus, a = 0 is always a solution. 0
Explanation
Problem Setup We are given the equation ( 8 1 ) − J a = 51 2 3 a and asked to solve for a . The possible solutions are a = − 8 , a = 0 , a = 8 , or no solution.
Rewriting with Common Base First, rewrite both sides of the equation with a common base. Note that 8 1 = 2 − 3 and 512 = 2 9 . Substituting these into the original equation, we get ( 2 − 3 ) − J a = ( 2 9 ) 3 a .
Simplifying Exponents Next, simplify the exponents. Using the property ( x m ) n = x mn , we have 2 3 J a = 2 27 a .
Equating Exponents Since the bases are equal, we can equate the exponents: 3 J a = 27 a .
Rearranging the Equation Rearrange the equation to isolate a : 3 J a − 27 a = 0 .
Factoring Factor out a : a ( 3 J − 27 ) = 0 .
Solving for a Now, solve for a . Either a = 0 or 3 J − 27 = 0 .
Analyzing the Solutions If 3 J − 27 = 0 , then 3 J = 27 , so J = 9 . In this case, any value of a satisfies the equation. If J = 9 , then the only solution is a = 0 .
Checking Possible Solutions If J = 9 , then any value of a is a solution. Let's check if a = − 8 or a = 8 are solutions when J = 9 . Substituting J = 9 into the original equation, we have ( 8 1 ) − 9 a = 51 2 3 a . If a = 0 , the equation becomes 1 = 1 , which is true. If a = − 8 , the left side is ( 8 1 ) − 9 ( − 8 ) = ( 8 1 ) 72 and the right side is 51 2 3 ( − 8 ) = 51 2 − 24 = ( 512 1 ) 24 = ( 2 9 1 ) 24 = ( 2 1 ) 216 = ( 8 1 ) 72 . Thus, a = − 8 is a solution when J = 9 . If a = 8 , the left side is ( 8 1 ) − 9 ( 8 ) = ( 8 1 ) − 72 = 8 72 and the right side is 51 2 3 ( 8 ) = 51 2 24 = ( 2 9 ) 24 = 2 216 = ( 2 3 ) 72 = 8 72 . Thus, a = 8 is a solution when J = 9 .
Final Answer If J = 9 , the only solution is a = 0 . If J = 9 , then a can be any real number. Since the possible solutions given are a = − 8 , a = 0 , a = 8 , or no solution, we consider two cases. If J = 9 , then a = − 8 , 0 , 8 are all solutions. If J = 9 , then a = 0 is the only solution. Without knowing the value of J , we can only say that a = 0 is a solution.
Examples
This problem demonstrates how exponential equations can be used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. By understanding how to solve these equations, we can make predictions about future outcomes based on current conditions. For example, in finance, we can use exponential equations to calculate the future value of an investment based on its initial principal, interest rate, and compounding frequency. Similarly, in environmental science, we can use exponential equations to model the decay of pollutants in the environment over time.
