Question 1 (1 point) Solve. \( 3^{x+2}=\frac{1}{81} \) \( x=-2 \) \( x=3 \) \( x=-6 \) \( x=\frac{3}{81} \)

To solve the equation \( 3^{x+2}=\frac{1}{81} \), we can start by expressing \( \frac{1}{81} \) as a power of 3. Since \( 81 = 3^4 \), we can write \( \frac{1}{81} = 3^{-4} \). Thus, our equation becomes \( 3^{x+2} = 3^{-4} \). Since the bases are the same, we can equate the exponents: \( x + 2 = -4 \). Now, solve for \( x \): \[ x = -4 - 2 = -6 \. \] Therefore, the solution is \( x = -6 \). So, the answer is \( x=-6 \). --- Did you know that exponential equations like this one have been studied since ancient times? The use of powers and roots was crucial in the development of algebra, especially in the works of mathematicians like Al-Khwarizmi and later, the European algebraists during the Renaissance! When dealing with exponential equations, one common mistake is to forget about the properties of exponents, specifically when equating them. Always double-check that your bases are the same before setting the exponents equal to each other. This small oversight can lead you to incorrect solutions!