Solve $6^{2 x+2}
\cdot 6^{3 x}=1$
$x=\square$
Answer (1)
Combine exponents on the left side: 6 2 x + 2 "." 6 3 x = 6 5 x + 2 .
Rewrite 1 as 6 0 , so the equation becomes 6 5 x + 2 = 6 0 .
Equate the exponents: 5 x + 2 = 0 .
Solve for x : x = − 5 2 .
− 5 2
Explanation
Understanding the Problem We are given the equation 6 2 x + 2 "." 6 3 x = 1 and need to solve for x .
Simplifying the Equation The base is the same, so we can add the exponents. We can rewrite 1 as 6 0 .
Combining Exponents Using the property a m "." a n = a m + n , we simplify the left side of the equation: 6 2 x + 2 "." 6 3 x = 6 ( 2 x + 2 ) + 3 x = 6 5 x + 2 So the equation becomes 6 5 x + 2 = 1 .
Expressing 1 as a Power of 6 Since 1 = 6 0 , we can rewrite the equation as 6 5 x + 2 = 6 0 .
Equating Exponents Since the bases are equal, we can equate the exponents: 5 x + 2 = 0 .
Solving for x Now, we solve for x :
5 x + 2 = 0 5 x = − 2 x = − 5 2 Therefore, x = − 5 2 = − 0.4 .
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, in finance, they are used to model compound interest. If you invest money in an account that pays compound interest, the amount of money you have after a certain period can be modeled using an exponential equation. Understanding how to solve these equations allows you to predict the growth of your investment over time.
