Then, multiply: \(9x^4y^6 \times 8x^3y^{12} = 72x^{4+3}y^{6+12} = 72x^7y^{18}\).
["Understanding Polynomial Multiplication: (9x^4y^6 \ imes 8x^3y^{12} = 72x^7y^{18})", "When working with algebra, multiplying monomials is a fundamental operation that forms the basis of more complex equations. One essential rule to remember is how to multiply like variables by combining like bases using the laws of exponents. In this article, we explore a key example: multiplying (9x^4y^6) and (8x^3y^{12}), resulting in (72x^7y^{18}).", "### The Multiplication Process Explained", "Start with the original expression:", "[\n9x^4y^6 \ imes 8x^3y^{12}\n]", "This involves multiplying coefficients and adding exponents of like variables.", "#### Step 1: Multiply the coefficients\nThe numerical coefficients 9 and 8 multiply simply as:", "[\n9 \ imes 8 = 72\n]", "#### Step 2: Apply the exponent rule (x^a \cdot x^b = x^{a+b})\nFor the variable (x):\n[\nx^4 \ imes x^3 = x^{4+3} = x^7\n]", "For the variable (y):\n[\ny^6 \ imes y^{12} = y^{6+12} = y^{18}\n]", "#### Final result:\nPutting it all together:", "[\n9x^4y^6 \ imes 8x^3y^{12} = 72x^7y^{18}\n]", "---", "### Why This Rule Works", "This simplification follows the fundamental laws of exponents, specifically the product of powers rule, which states that when multiplying powers with the same base, you add the exponents. This rule applies to any variable with variable exponents and ensures consistency in algebraic expressions.", "---", "### Why This Matters in Algebra and Beyond", "Understanding how to multiply polynomials and apply exponent rules is crucial in solving equations, simplifying expressions, and preparing for higher-level math like calculus and linear algebra. This basic operation becomes a building block for more complex manipulations, including solving systems of equations, working with functions, and designing mathematical models.", "---", "### Practice Tip", "Try multiplying your own expressions like:\n[\n4x^2y^3 \ imes 5x^4y^5 \quad \ ext{or} \quad 3a^2b \ imes 2a^3b^4\n]\nto reinforce your understanding of combining like terms and exponent addition.", "---", "Conclusion:", "The multiplication rule (9x^4y^6 \ imes 8x^3y^{12} = 72x^7y^{18}) might look simple, but it illustrates a powerful concept in algebra: combining coefficients and exponents efficiently. Mastery of this principle is key to tackling more advanced math challenges with confidence.", "---", "Keywords:\npolynomial multiplication, exponent rules, multiplying monomials, algebra examples, (9x^4y^6 \ imes 8x^3y^{12} = 72x^7y^{18}), coefficient multiplication, law of exponents, algebra fundamentals.\nMeta Description:\nLearn how to multiply (9x^4y^6 \ imes 8x^3y^{12}) using exponent rules to get (72x^7y^{18}). Understand the steps behind polynomial multiplication and exponent addition."]
