Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( z = -20 \)
Solution
Para resolver la ecuación \( 8 - (3z - 4)^{1/3} = 12 \), sigue estos pasos:
1. **Aislar el término con la raíz cúbica:**
\[
8 - (3z - 4)^{1/3} = 12
\]
Restamos 8 de ambos lados:
\[
-(3z - 4)^{1/3} = 12 - 8
\]
\[
-(3z - 4)^{1/3} = 4
\]
2. **Eliminar el signo negativo:**
Multiplicamos ambos lados por -1 para deshacernos del signo negativo:
\[
(3z - 4)^{1/3} = -4
\]
3. **Elevar al cubo para eliminar la raíz cúbica:**
\[
\left( (3z - 4)^{1/3} \right)^3 = (-4)^3
\]
\[
3z - 4 = -64
\]
4. **Resolver para \( z \):**
Sumamos 4 a ambos lados:
\[
3z = -64 + 4
\]
\[
3z = -60
\]
Dividimos entre 3:
\[
z = \frac{-60}{3}
\]
\[
z = -20
\]
**Respuesta final:** \( z = -20 \)
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Simplify this solution Extra Insights
To solve \( 8-(3 z-4)^{1 / 3}=12 \), we can start by isolating the cube root term. First, subtract 8 from both sides: \[ -(3z-4)^{1/3} = 12 - 8 \] \[ -(3z-4)^{1/3} = 4 \] Now, multiply both sides by -1: \[ (3z-4)^{1/3} = -4 \] Next, we will cube both sides to eliminate the cube root: \[ 3z - 4 = (-4)^3 \] \[ 3z - 4 = -64 \] Now, add 4 to both sides: \[ 3z = -64 + 4 \] \[ 3z = -60 \] Finally, divide by 3: \[ z = -20 \] So, the solution is \( z = -20 \).
