Jasonbradley
Home

((81)^n*3^5-(3)^(4n-1)*(243))/((9^(2n))(3^3))

4 min read Jun 16, 2024
((81)^n*3^5-(3)^(4n-1)*(243))/((9^(2n))(3^3))

Simplifying the Expression: ((81)^n3^5-(3)^(4n-1)(243))/((9^(2n))(3^3))

This article will guide you through simplifying the complex mathematical expression: ((81)^n3^5-(3)^(4n-1)(243))/((9^(2n))(3^3))

Understanding the Basics

Before diving into the simplification, let's break down the components:

  • Exponents: The expression involves exponents, which represent repeated multiplication. For example, 3^5 means 3 multiplied by itself five times (3 * 3 * 3 * 3 * 3).
  • Base and Power: In an exponent, the base is the number being multiplied, and the power indicates the number of times it's multiplied by itself. In 3^5, 3 is the base and 5 is the power.

Step-by-Step Simplification

  1. Express all terms with base 3:

    • 81 can be written as 3^4 (since 3 * 3 * 3 * 3 = 81)
    • 243 can be written as 3^5 (since 3 * 3 * 3 * 3 * 3 = 243)
    • 9 can be written as 3^2 (since 3 * 3 = 9)

    So, our expression becomes: ((3^(4n)3^5-(3)^(4n-1)(3^5))/((3^(2*2n))(3^3))

  2. Apply exponent rules:

    • Product of powers: When multiplying exponents with the same base, you add the powers. For example, 3^4 * 3^5 = 3^(4+5) = 3^9.
    • Power of a power: When raising a power to another power, you multiply the powers. For example, (3^2)^3 = 3^(2*3) = 3^6.

    Using these rules, we can simplify the expression: (3^(4n+5) - 3^(4n-1+5))/(3^(4n+3))

  3. Further simplification:

    • Combine like terms: (3^(4n+5) - 3^(4n+4))/(3^(4n+3))

  4. Factoring out common factors:

    • Notice that both the numerator and denominator share a common factor of 3^(4n+3): (3^(4n+3) * (3^2 - 3^1))/(3^(4n+3))

  5. Canceling out common factors:

    • Since the expression has the same factor in the numerator and denominator, they cancel out: (3^2 - 3^1)

  6. Final calculation:

    • Simplify: (9 - 3) = 6

Conclusion

By applying exponent rules and factoring, we successfully simplified the expression ((81)^n3^5-(3)^(4n-1)(243))/((9^(2n))(3^3)) to a simple constant: 6.

Featured Posts