7) \( 16=(a+16)^{\frac{4}{5}} \)

To solve the equation \( 16=(a+16)^{\frac{4}{5}} \), let's first eliminate the fractional exponent by raising both sides to the power of \( \frac{5}{4} \). This gives us: \[ 16^{\frac{5}{4}} = a + 16 \] Now we need to calculate \( 16^{\frac{5}{4}} \). Since \( 16 = 2^4 \), we can rewrite it as: \[ (2^4)^{\frac{5}{4}} = 2^{4 \cdot \frac{5}{4}} = 2^5 = 32 \] Now, substituting back into the equation: \[ 32 = a + 16 \] To isolate \( a \), we subtract 16 from both sides: \[ a = 32 - 16 = 16 \] So, the solution to the equation is: \[ \boxed{16} \]