(x^2/5 Y^4/10)^-5

Jasonbradley

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Jun 17, 2024

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Simplifying (x^2/5 y^4/10)^-5

This problem involves simplifying an expression with negative exponents and fractions. Let's break down the steps:

Understanding the Rules

• Negative Exponents: A term raised to a negative exponent is equal to its reciprocal raised to the positive version of that exponent.
  • Example: x^-2 = 1/x^2
• Fractional Exponents: When a fraction is raised to an exponent, both the numerator and denominator are raised to that exponent.
  • Example: (a/b)^n = a^n/b^n
• Power of a Power: When a power is raised to another power, the exponents are multiplied.
  • Example: (x^m)^n = x^(m*n)

Applying the Rules

1. Reciprocal: Since the expression is raised to a negative exponent, we take its reciprocal and change the exponent to positive:

   • (x^2/5 y^4/10)^-5 = (1 / (x^2/5 y^4/10))^5

2. Simplifying the Reciprocal:

   • (1 / (x^2/5 y^4/10))^5 = (5y^4/10 / x^2)^5

3. Power of a Power: Apply the power of a power rule to both the numerator and the denominator:

   • (5y^4/10 / x^2)^5 = (5^5 * y^(45) / 10^5) / x^(25)

4. Simplifying Exponents: Calculate the exponents:

   • (5^5 * y^(45) / 10^5) / x^(25) = (3125 y^20 / 100000) / x^10

5. Final Simplification: Combine the constants:

   • (3125 y^20 / 100000) / x^10 = (125 y^20) / (400 x^10)

The Result

The simplified form of (x^2/5 y^4/10)^-5 is (125 y^20) / (400 x^10).