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(5/x)^-2 X (25/3x)^7 (5/2x)^-4

2 min read Jun 16, 2024
(5/x)^-2 X (25/3x)^7 (5/2x)^-4

Simplifying Expressions with Exponents

This article will guide you through simplifying the expression: (5/x)^-2 x (25/3x)^7 x (5/2x)^-4. We'll break down the process step-by-step, utilizing the rules of exponents.

Understanding the Rules of Exponents

Before diving into the simplification, let's refresh our memory on some key exponent rules:

  • Negative Exponents: a^-n = 1/a^n
  • Fractional Exponents: a^(m/n) = (a^m)^(1/n) = (a^(1/n))^m
  • Product of Powers: a^m * a^n = a^(m+n)
  • Power of a Power: (a^m)^n = a^(m*n)
  • Power of a Quotient: (a/b)^n = a^n / b^n

Step-by-Step Simplification

  1. Dealing with Negative Exponents:

    • (5/x)^-2 = (x/5)^2
    • (5/2x)^-4 = (2x/5)^4

  2. Simplifying using Power of a Quotient:

    • (x/5)^2 = x^2 / 5^2 = x^2 / 25
    • (2x/5)^4 = (2x)^4 / 5^4 = 16x^4 / 625

  3. Combining terms:

    • (x^2 / 25) x (25/3x)^7 x (16x^4 / 625)

  4. Applying Power of a Power:

    • (25/3x)^7 = 25^7 / (3x)^7 = 78125 / 2187x^7

  5. Combining terms again:

    • (x^2 / 25) x (78125 / 2187x^7) x (16x^4 / 625)

  6. Multiplying fractions:

    • (x^2 * 78125 * 16x^4) / (25 * 2187x^7 * 625)

  7. Simplifying by canceling common factors:

    • (16 * 78125 * x^6) / (25 * 2187 * 625 * x^7)

    • 1024 / 3375x

Final Result

The simplified expression is 1024 / 3375x.

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