(3d^-4n^5)^-3/12d^3n^4

Jasonbradley

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Jun 16, 2024

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Simplifying Expressions with Exponents

This article will guide you through the process of simplifying the expression (3d^-4n^5)^-3 / 12d^3n^4. We will use the rules of exponents to break down the problem step by step.

Understanding the Rules of Exponents

Before we begin, let's review some key 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
• Negative Exponent: x^-n = 1/x^n

Simplifying the Expression

Let's break down the simplification step by step:

1. Simplify the numerator:

   (3d^-4n^5)^-3 = 3^-3 * (d^-4)^-3 * (n^5)^-3

   Applying the power of a power rule, we get:

   = 3^-3 * d^12 * n^-15

2. Apply the negative exponent rule to the numerator:

   = 1/3^3 * d^12 * 1/n^15

   = d^12 / (3^3 * n^15)

3. Combine the numerator and denominator:

   = d^12 / (27n^15)

4. Simplify the denominator:

   = d^12 / (27n^15)

5. Simplify the entire expression:

   (3d^-4n^5)^-3 / 12d^3n^4 = (d^12 / (27n^15)) / (12d^3n^4)

   Applying the quotient of powers rule, we get:

   = d^(12-3) / (27 * 12 * n^(15+4))

   = d^9 / (324n^19)

Conclusion

Therefore, the simplified form of the expression (3d^-4n^5)^-3 / 12d^3n^4 is d^9 / (324n^19). This process illustrates how understanding the rules of exponents allows us to simplify complex expressions into a more manageable form.