Use the like bases property to solve the equation \( \left(\frac{1}{3}\right)^{n}=243 \)

To solve the equation \( \left(\frac{1}{3}\right)^{n} = 243 \), we first need to express both sides using the same base. We know that \( 243 \) can be written as \( 3^5 \) because \( 3 \times 3 \times 3 \times 3 \times 3 = 243 \). We also note that \( \frac{1}{3} \) is the same as \( 3^{-1} \). So we rewrite the left side: \[ \left(\frac{1}{3}\right)^{n} = (3^{-1})^{n} = 3^{-n} \] Now, replacing \( 243 \) with \( 3^5 \) gives us: \[ 3^{-n} = 3^5 \] Since the bases are the same, we can set the exponents equal to each other: \[ -n = 5 \] To solve for \( n \), multiply both sides by -1: \[ n = -5 \] Thus, the solution to the equation \( \left(\frac{1}{3}\right)^{n} = 243 \) is \( n = -5 \)!