%5E2.jpeg "(2x^5y^8/5x^6y^5)^2")
Simplifying the Expression: (2x⁵y⁸/5x⁶y⁵)²
This article will explore the simplification of the expression (2x⁵y⁸/5x⁶y⁵)². We will break down the process step by step to understand how to handle exponents within fractions.
Understanding the Rules
Before we dive into the simplification, let's recall some fundamental exponent rules:
- Product of powers: xᵃ * xᵇ = xᵃ⁺ᵇ
- Quotient of powers: xᵃ / xᵇ = xᵃ⁻ᵇ
- Power of a power: (xᵃ)ᵇ = xᵃᵇ
- Power of a product: (xy)ᵃ = xᵃyᵃ
- Power of a quotient: (x/y)ᵃ = xᵃ/yᵃ
Step-by-Step Simplification
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Apply the power of a quotient rule: (2x⁵y⁸/5x⁶y⁵)² = (2²x⁵²y⁸²/5²x⁶²y⁵²)
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Simplify the powers: (2²x⁵²y⁸²/5²x⁶²y⁵²) = (4x¹⁰y¹⁶/25x¹²y¹⁰)
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Apply the quotient of powers rule: (4x¹⁰y¹⁶/25x¹²y¹⁰) = (4/25) * x¹⁰⁻¹² * y¹⁶⁻¹⁰
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Simplify the exponents: (4/25) * x¹⁰⁻¹² * y¹⁶⁻¹⁰ = (4/25) * x⁻² * y⁶
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Express negative exponents in the denominator: (4/25) * x⁻² * y⁶ = (4y⁶)/(25x²)
Conclusion
Therefore, the simplified form of the expression (2x⁵y⁸/5x⁶y⁵)² is (4y⁶)/(25x²). By carefully applying the exponent rules, we successfully reduced the complex expression into a more manageable form.