Pregunta
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Respuesta verificada por el tutor
Responder
\[
\left(x^{3} - 3 y^{6}\right)^{2} = x^6 - 6x^3 y^6 + 9y^{12}
\]
Solución
To expand the expression \( \left(x^{3} - 3 y^{6}\right)^{2} \), you can use the algebraic identity for squaring a binomial:
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
**Step-by-Step Expansion:**
1. **Identify \( a \) and \( b \):**
\[
a = x^3 \quad \text{and} \quad b = 3y^6
\]
2. **Apply the Binomial Formula:**
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
3. **Compute Each Term:**
- **First Term (\( a^2 \)):**
\[
(x^3)^2 = x^{3 \times 2} = x^6
\]
- **Second Term (\( -2ab \)):**
\[
-2 \times x^3 \times 3y^6 = -6x^3 y^6
\]
- **Third Term (\( b^2 \)):**
\[
(3y^6)^2 = 9y^{6 \times 2} = 9y^{12}
\]
4. **Combine All Terms:**
\[
x^6 - 6x^3 y^6 + 9y^{12}
\]
**Final Expanded Form:**
\[
\left(x^{3} - 3 y^{6}\right)^{2} = x^6 - 6x^3 y^6 + 9y^{12}
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Mind Expander
Let's dive into expanding this expression! When you square the binomial \( \left(x^{3} - 3 y^{6}\right)^{2} \), you can use the binomial theorem or FOIL method. The result will be \( x^{6} - 2 \cdot x^{3} \cdot 3y^{6} + (3y^{6})^{2} \), leading to \( x^{6} - 6x^{3}y^{6} + 9y^{12} \). Voilà! Now, not only do you have the expanded form, but it's also a lovely polynomial that can take on lots of different values based on the inputs for \(x\) and \(y\). It showcases the elegant interplay of variables and coefficients!
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