["# Simplify the Expression: (2x²y³)³ × (3xy²)² — A Step-by-Step Guide", "Algebraic expressions often look intimidating at first, but with the right approach, simplifying complex expressions becomes straightforward. In this article, we will learn how to simplify the expression (2x²y³)³ × (3xy²)² efficiently using exponent rules and multiplication principles. Whether you're a student mastering algebra or someone brushing up on fundamental math concepts, this guide breaks it down clearly.", "---", "### The Expression to Simplify", "We are simplifying:
\n$$
\n(2x^2y^3)^3 \ imes (3xy^2)^2
\n$$", "---", "### Step 1: Apply the Power of a Product Rule", "The first key rule we use is:
\n$$
\n(a \ imes b)^n = a^n \ imes b^n
\n$$
\nThis means we can apply exponents to each factor inside parentheses when raising an entire product to a power.", "So rewrite the expression by distributing the exponents:", "$$
\n(2x^2y^3)^3 = 2^3 \ imes (x^2)^3 \ imes (y^3)^3
\n$$
\n$$
\n(3xy^2)^2 = 3^2 \ imes x^2 \ imes (y^2)^2
\n$$", "---", "### Step 2: Compute Each Term", "Now compute the numerical and variable parts separately.", "Numerical parts:", "- $ 2^3 = 8 $
\n- $ 3^2 = 9 $", "Variable parts using power of power rule: $ (a^m)^n = a^{mn} $", "- $ (x^2)^3 = x^{2 \ imes 3} = x^6 $
\n- $ (y^3)^3 = y^{3 \ imes 3} = y^9 $
\n- $ x^2 $ remains as is
\n- $ (y^2)^2 = y^{2 \ imes 2} = y^4 $", "Now rewrite the expression with these simplified powers:", "$$
\n8 \cdot x^6 \cdot y^9 \ imes 9 \cdot x^2 \cdot y^4
\n$$", "---", "### Step 3: Multiply the Constants and Combine Like Terms", "Multiply constants first:", "$$
\n8 \ imes 9 = 72
\n$$", "Now collect like variables using multiplication rules: $ a^m \ imes a^n = a^{m+n} $", "- $ x^6 \ imes x^2 = x^{6+2} = x^8 $
\n- $ y^9 \ imes y^4 = y^{9+4} = y^{13} $", "---", "### Final Simplified Expression", "Putting it all together:", "$$
\n72x^8y^{13}
\n$$", "---", "### Why This Method Works", "By applying exponent rules—especially distributing exponents over multiplication and using power of a power identity—we avoid expanding into lengthy polynomials. This simplification is faster, less error-prone, and preserves the expression’s meaning.", "---", "### Conclusion", "Simplifying expressions like (2x²y³)³ × (3xy²)² becomes clear and manageable by breaking the problem into steps: applying exponent rules, expanding powers, multiplying constants, and combining like terms. Mastering this approach builds a strong foundation for higher-level algebra, calculus, and beyond.", "Key Takeaways:", "- Use $ (ab)^n = a^n b^n $
\n- Use $ (a^m)^n = a^{mn} $
\n- Combine like terms after expansion", "Now you can confidently simplify complex algebraic expressions with ease!", "---", "### Related Keywords for SEO Optimization", "- Simplify algebraic expressions
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