Solve: [tex]36^{3 s}=216^{2 s+1}[/tex]
Answer (1)
∙ Express 36 and 216 as powers of 6 : 36 = 6 2 and 216 = 6 3 .$\bullet S u b s t i t u t e in t o t h ee q u a t i o n : (6^2)^{3s} = (6^3)^{2s+1} .$ ∙ Simplify the exponents: 6 6 s = 6 6 s + 3 .$\bullet Eq u a t e t h ee x p o n e n t s an d so l v e f or s$: 6 s = 6 s + 3 ⟹ 0 = 3 , which is a contradiction. Therefore, the equation has no solution .
Explanation
Understanding the Problem We are given the equation 3 6 3 s = 21 6 2 s + 1 and need to solve for s .
Expressing with a Common Base Notice that both 36 and 216 can be expressed as powers of 6. We have 36 = 6 2 and 216 = 6 3 .
Substitution Substituting these into the original equation, we get ( 6 2 ) 3 s = ( 6 3 ) 2 s + 1 .
Simplifying Exponents Using the power of a power rule, we simplify the exponents: 6 6 s = 6 3 ( 2 s + 1 ) , which gives 6 6 s = 6 6 s + 3 .
Equating Exponents Since the bases are equal, we can equate the exponents: 6 s = 6 s + 3 .
Solving for s Subtracting 6 s from both sides, we get 0 = 3 , which is a contradiction. This means there is no solution for s that satisfies the equation.
Final Answer Therefore, the equation 3 6 3 s = 21 6 2 s + 1 has no solution.
Examples
Exponential equations like this are used in various fields, such as calculating population growth, radioactive decay, and compound interest. For instance, if a bacterial population triples every hour, we can use exponential equations to predict its size after a certain number of hours. Similarly, in finance, compound interest calculations rely on exponential growth to determine the future value of an investment.
