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(243)^n/5*3^2 N+1/9^n*3^n-1

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

Simplifying the Expression (243)^n/53^2n+1/9^n3^n-1

This article explores the simplification of the expression (243)^n/53^2n+1/9^n3^n-1. We will break down the expression step-by-step, utilizing the laws of exponents to achieve a concise and simplified form.

Understanding the Expression

The expression consists of various powers of 3 and 9, along with constant terms. To simplify it, we need to leverage 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)

Simplification Steps

  1. Express all terms as powers of 3:

    • 243 = 3^5, so (243)^n = (3^5)^n = 3^(5n)
    • 9 = 3^2, so 9^n = (3^2)^n = 3^(2n)

  2. Substitute the simplified terms back into the expression:

    • (243)^n/53^2n+1/9^n3^n-1 = (3^(5n))/5 * 3^(2n+1) / (3^(2n)) * 3^(n-1)

  3. Combine terms with the same base using the product and quotient of powers rules:

    • (3^(5n)) * 3^(2n+1) * 3^(n-1) / (5 * 3^(2n)) = 3^(5n + 2n + 1 + n - 1) / (5 * 3^(2n)) = 3^(8n) / (5 * 3^(2n))

  4. Simplify further using the quotient of powers rule:

    • 3^(8n) / (5 * 3^(2n)) = 3^(8n-2n) / 5 = 3^(6n) / 5

Final Result

The simplified form of the expression (243)^n/53^2n+1/9^n3^n-1 is 3^(6n) / 5. This simplified form is easier to work with and understand, especially when performing further operations or calculations.

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