Alright, let's solve the equation:
[tex]\[ -8 = -4 + (q-3)^{1/3} \][/tex]
Step-by-step:
1. Isolate the term involving [tex]\( q \)[/tex]:
[tex]\[ -8 + 4 = (q-3)^{1/3} \][/tex]
This simplifies to:
[tex]\[ -4 = (q-3)^{1/3} \][/tex]
2. Eliminate the cube root:
To eliminate the cube root, we will raise both sides of the equation to the power of 3:
[tex]\[ (-4)^3 = \left((q-3)^{1/3}\right)^3 \][/tex]
3. Simplify the powers:
[tex]\[ -64 = q - 3 \][/tex]
4. Solve for [tex]\( q \)[/tex]:
[tex]\[ q - 3 = -64 \][/tex]
Add 3 to both sides:
[tex]\[ q = -64 + 3 \][/tex]
[tex]\[ q = -61 \][/tex]
Thus, the solution to the equation [tex]\( -8 = -4 + (q-3)^{1/3} \)[/tex] is [tex]\( q = -61 \)[/tex].
Given that there was no solution found mathematically, this indicates there might have been no valid value of [tex]\( q \)[/tex] that satisfies the original equation within the real numbers. This implies there could be an error or special condition involved; however, based on our standard algebraic steps, this is our logical outcome.
