(5xy%5E-8)%2F(x%5E-3)%5E4y%5E-2.jpeg "(3x^-2y^3)(5xy^-8)/(x^-3)^4y^-2")
Simplifying Algebraic Expressions: A Step-by-Step Guide
This article will guide you through the process of simplifying the algebraic expression: (3x^-2y^3)(5xy^-8)/(x^-3)^4y^-2. We'll break down the steps and use the rules of exponents to arrive at a simplified solution.
Understanding the Rules of Exponents
Before we begin, let's refresh our memory on some key exponent rules:
- Product of Powers: x^m * x^n = x^(m+n)
- Quotient of Powers: x^m / x^n = x^(m-n)
- Power of a Power: (x^m)^n = x^(m*n)
- Negative Exponent: x^-n = 1/x^n
Simplifying the Expression
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Distribute: Begin by distributing the multiplication in the numerator: (3x^-2y^3)(5xy^-8) = 15x^(-2+1)y^(3-8) = 15x^-1y^-5
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Simplify the Denominator: Apply the "Power of a Power" rule to the denominator: (x^-3)^4 = x^(-3*4) = x^-12
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Combine the Numerator and Denominator: Now we have: (15x^-1y^-5) / (x^-12y^-2)
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Apply the Quotient of Powers Rule:
15x^(-1 - (-12))y^(-5 - (-2)) = 15x^11y^-3 -
Express with Positive Exponents:
15x^11 / y^3
Final Simplified Expression
The simplified form of the given expression is 15x^11 / y^3.
Key Takeaways
Simplifying algebraic expressions can seem daunting, but by understanding the rules of exponents and breaking down the problem into smaller steps, it becomes a manageable process. Remember to always pay attention to the signs of the exponents and apply the appropriate rules.