(2x^-3y^-2)^5/(6x^-1 Y^-8)^2

Jasonbradley

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Jun 16, 2024

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Simplifying Algebraic Expressions: (2x^-3y^-2)^5/(6x^-1 y^-8)^2

This article will guide you through the process of simplifying the algebraic expression: (2x^-3y^-2)^5/(6x^-1 y^-8)^2. Let's break down the steps involved.

Understanding the Rules

To simplify this expression, we'll use the following 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)
• Power of a Product: (x*y)^n = x^n * y^n
• Power of a Quotient: (x/y)^n = x^n / y^n

Step-by-Step Simplification

1. Apply the Power of a Power rule:

   • (2x^-3y^-2)^5 = 2^5 * x^(-35) * y^(-25) = 32x^-15y^-10
   • (6x^-1y^-8)^2 = 6^2 * x^(-12) * y^(-82) = 36x^-2y^-16

2. Substitute the simplified terms back into the original expression:

   • (32x^-15y^-10) / (36x^-2y^-16)

3. Apply the Quotient of Powers rule:

   • 32/36 * x^(-15 - (-2)) * y^(-10 - (-16))

4. Simplify the coefficients and exponents:

   • (8/9) * x^-13 * y^6

5. Rewrite negative exponents using the rule x^-n = 1/x^n:

   • (8/9) * (1/x^13) * y^6

6. Combine terms:

   • 8y^6 / (9x^13)

Final Result

The simplified form of the expression (2x^-3y^-2)^5/(6x^-1 y^-8)^2 is 8y^6 / (9x^13).