(3t^-4 N^-5)^-3/4n^3

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (3t^-4 n^-5)^-3/4n^3

This expression involves exponents and negative exponents. To simplify it, 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)
• Negative exponent: x^-n = 1/x^n

Let's break down the simplification step by step:

1. Simplify the inside of the parentheses:

   • (3t^-4 n^-5)^-3 = (3 * t^-4 * n^-5)^-3
   • Applying the negative exponent rule, we get: (3 * 1/t^4 * 1/n^5)^-3 = (3/t^4n^5)^-3

2. Apply the power of a power rule:

   • (3/t^4n^5)^-3 = 3^-3 / (t^4n^5)^-3 = 3^-3 / t^-12n^-15

3. Apply the negative exponent rule again:

   • 3^-3 / t^-12n^-15 = 1/3^3 * t^12 * n^15 = t^12 * n^15 / 3^3

4. Combine with the remaining term:

   • (t^12 * n^15 / 3^3) * 4n^3 = (4 * t^12 * n^15 * n^3) / 3^3

5. Simplify using the product of powers rule:

   • (4 * t^12 * n^(15+3)) / 3^3 = (4 * t^12 * n^18) / 27

Therefore, the simplified expression is (4 * t^12 * n^18) / 27.