%5E-3.jpeg "(10x^3y^2/5x^-3y^4)^-3")
Simplifying Exponential Expressions: (10x^3y^2/5x^-3y^4)^-3
This article will guide you through the process of simplifying the expression (10x^3y^2/5x^-3y^4)^-3.
Understanding the Rules
Before we dive into the simplification, let's recall some key rules of exponents:
- 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
Step-by-Step Simplification
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Simplify inside the parentheses:
- Divide the coefficients: 10/5 = 2
- Apply the quotient of powers rule for x: x^(3-(-3)) = x^6
- Apply the quotient of powers rule for y: y^(2-4) = y^-2
- The simplified expression inside the parentheses is: 2x^6y^-2
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Apply the power of a power rule:
- (2x^6y^-2)^-3 = 2^-3 * (x^6)^-3 * (y^-2)^-3
- Simplify: 2^-3 * x^-18 * y^6
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Simplify negative exponents:
- 2^-3 = 1/2^3 = 1/8
- x^-18 = 1/x^18
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Combine the terms:
- The final simplified expression is 1/(8x^18) * y^6 or y^6/(8x^18)
Conclusion
By applying the rules of exponents systematically, we simplified the expression (10x^3y^2/5x^-3y^4)^-3 to y^6/(8x^18). This process illustrates how understanding these rules allows you to manipulate complex expressions and express them in a more manageable form.