Use the like bases property to solve the equation \( \left(\frac{1}{2}\right)^{x+2}=2^{4 x+5} \) \( x=\square \) Question Help: \( \square \) Video

To solve the equation \( \left(\frac{1}{2}\right)^{x+2}=2^{4 x+5} \), we can express \( \frac{1}{2} \) as \( 2^{-1} \). This rewrites our equation to \( (2^{-1})^{x+2}=2^{4x+5} \). Applying the power of a power property gives us \( 2^{-(x+2)}=2^{4x+5} \). Since the bases are the same, we set the exponents equal: \(-1(x+2) = 4x + 5\). Solving this gives us \( -x - 2 = 4x + 5 \), which rearranges to \( -5x = 7 \), leading to \( x = -\frac{7}{5} \). For this equation, always remember to keep track of your signs while manipulating exponents! A common mistake is to forget that when you multiply powers, you add the exponents, and when you have negative bases or negative exponents, things can get a bit tricky!