Question: Compute the square of $(2x^2 - 3y)^3$ and simplify the expression.
["Perfectly Computing the Square of $(2x^2 - 3y)^3$: Simplified Expression and Step-By-Step Guide", "When tackling algebraic expressions like the square of $(2x^2 - 3y)^3$, careful computation and simplification are essential—especially if you're preparing for calculus, advanced algebra exams, or applied math problems. In this SEO-optimized article, we’ll guide you step-by-step through computing the square of $(2x^2 - 3y)^3$, simplify the result, and explain how to approach similar expressions with confidence.", "---", "### Understanding the Expression: Why Square It?", "We want to compute:", "$$\n\left[(2x^2 - 3y)^3\right]^2\n$$", "This is equivalent to raising $(2x^2 - 3y)^3$ to the power of 2, which simplifies directly to:", "$$\n(2x^2 - 3y)^6\n$$", "So instead of computing a square of a power, we compute the 6th power of the binomial. But why go this far? Knowing the simplified form helps in integration, differentiation, and modeling in physics or engineering.", "---", "### Step 1: Apply the Power of a Binomial Formula", "Recall the binomial expansion formula:", "$$\n(a - b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} (-b)^k\n$$", "Here, $ a = 2x^2 $, $ b = 3y $, and $ n = 6 $. So:", "$$\n(2x^2 - 3y)^6 = \sum_{k=0}^{6} \binom{6}{k} (2x^2)^{6-k} (-3y)^k\n$$", "---", "### Step 2: Expand Each Term", "Compute each term using binomial coefficients and exponents:", "- Binomial coefficients for $n = 6$:\n $\binom{6}{0} = 1$, $\binom{6}{1} = 6$, $\binom{6}{2} = 15$, $\binom{6}{3} = 20$, $\binom{6}{4} = 15$, $\binom{6}{5} = 6$, $\binom{6}{6} = 1$", "- Powers:\n $(2x^2)^{6-k} = 2^{6-k} x^{2(6-k)} = 64x^{12 - 2k}$\n $(-3y)^k = (-3)^k y^k$", "Now write each term:", "$$\n(2x^2 - 3y)^6 = \binom{6}{0}(64x^{12})(-1)^0(y)^0 + \binom{6}{1}(64x^{10})(-3)^1 y^1 + \binom{6}{2}(64x^8)(-3)^2 y^2 + \cdots + (2x^2 - 3y)^6\n$$", "Expanding all terms:", "- $k = 0$: $1 \cdot 64x^{12} \cdot 1 = 64x^{12}$\n- $k = 1$: $6 \cdot 64x^{10} \cdot (-3) = -1152x^{10}y$\n- $k = 2$: $15 \cdot 64x^8 \cdot 9y^2 = 8640x^8y^2$\n- $k = 3$: $20 \cdot 64x^6 \cdot (-27)y^3 = -34560x^6y^3$\n- $k = 4$: $15 \cdot 64x^4 \cdot 81y^4 = 77760x^4y^4$\n- $k = 5$: $6 \cdot 64x^2 \cdot (-243)y^5 = -93312x^2y^5$\n- $k = 6$: $1 \cdot 64x^0 \cdot 729y^6 = 729y^6$", "---", "### Step 3: Combine All Terms", "Now add all contributions:", "$$\n(2x^2 - 3y)^6 = 64x^{12} - 1152x^{10}y + 8640x^8y^2 - 34560x^6y^3 + 77760x^4y^4 - 93312x^2y^5 + 729y^6\n$$", "---", "### Final Simplified Expression", "$$\n\boxed{(2x^2 - 3y)^6 = 64x^{12} - 1152x^{10}y + 8640x^8y^2 - 34,!560x^6y^3 + 77,!760x^4y^4 - 93,!312x^2y^5 + 729y^6}\n$$", "---", "### Why This Simplification Matters", "Simplifying algebraic expressions like this:", "- Makes integration and differentiation easier in calculus\n- Clarifies terms in polynomial modeling\n- Improves performance in computational algorithms\n- Strengthens problem-solving skills for advanced STEM courses", "---", "### Pro Tips for Computing Powers of Complex Binomials", "- Always apply the binomial theorem systematically\n- Keep track of coefficients using Pascal’s triangle or binomial coefficients\n- Simplify exponents and signs early to avoid errors\n- Check for common factors across terms if possible\n- Consider using symbolic computation tools (like Python with sympy) to verify large expressions", "---", "### Related Keywords for SEO Optimization:", "- Compute $(2x^2 - 3y)^3$ square\n- Simplify $(2x^2 - 3y)^6$\n- Binomial expansion $(2x^2 - 3y)^6`\n- Algebraic expressions simplified\n- How to expand $(a - b)^n$\n- Step-by-step binomial power calculation\n- Algebraic identities and formulas", "---", "By mastering expressions like $\left[(2x^2 - 3y)^3\right]^2$, you build critical algebraic fluency essential for advanced math, science, and engineering applications.\nKeep practicing with powers and binomials—confidence grows with each step!"]
