(4mn/m^-2n^6)^-2

Jasonbradley

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Jun 16, 2024

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Simplifying Expressions with Negative Exponents

This article will guide you through simplifying the expression (4mn/m^-2n^6)^-2. We'll break down the process step-by-step, using the rules of exponents.

Understanding Negative Exponents

The key to solving this problem lies in understanding how negative exponents work. A negative exponent indicates a reciprocal. This means:

• x^-n = 1/x^n

Simplifying the Expression

1. Apply the Power of a Quotient Rule: This rule states that the power of a quotient is equal to the quotient of the powers of the numerator and denominator.
   (a/b)^n = a^n / b^n

   Applying this to our expression: (4mn/m^-2n^6)^-2 = (4^-2m^-2n^-2) / (m^4n^-12)

2. Simplify using Negative Exponent Rule: We'll now apply the rule for negative exponents to each term. (4^-2m^-2n^-2) / (m^4n^-12) = (1/4^2 * 1/m^2 * 1/n^2) / (m^4 * 1/n^12)

3. Combine Like Terms: To simplify further, we'll combine the terms with the same base. (1/16 * 1/m^2 * 1/n^2) / (m^4 * 1/n^12) = (1/16m^2n^2) / (m^4/n^12)

4. Dividing by a Fraction: Remember that dividing by a fraction is the same as multiplying by its reciprocal. (1/16m^2n^2) / (m^4/n^12) = (1/16m^2n^2) * (n^12/m^4)

5. Multiplying Fractions: We can now multiply the numerators and denominators. (1/16m^2n^2) * (n^12/m^4) = n^12 / (16m^6n^2)

6. Simplifying Further: Finally, we simplify by subtracting the exponents of like bases. n^12 / (16m^6n^2) = n^10 / 16m^6

Final Result

Therefore, the simplified form of (4mn/m^-2n^6)^-2 is n^10 / 16m^6.