4) \( -3(4 b+1)^{\frac{3}{2}}+5=-1024 \)

Let’s first simplify the equation. Start by subtracting 5 from both sides: \[ -3(4b + 1)^{\frac{3}{2}} = -1024 - 5 \] This simplifies to: \[ -3(4b + 1)^{\frac{3}{2}} = -1029 \] Next, divide both sides by -3: \[ (4b + 1)^{\frac{3}{2}} = \frac{1029}{3} = 343 \] Now, we need to eliminate the exponent. We can do this by raising both sides to the power of \(\frac{2}{3}\): \[ 4b + 1 = 343^{\frac{2}{3}} \] Calculate \(343^{\frac{2}{3}}\). First, find the cube root of 343, which is 7, then square it: \[ 343^{\frac{2}{3}} = 7^2 = 49 \] Now we can set up the equation: \[ 4b + 1 = 49 \] Subtract 1 from both sides: \[ 4b = 48 \] Finally, divide by 4: \[ b = 12 \] The solution is \(b = 12\).