(-2a^3b^2c^0/3a^2b^3c^7)^-2

Jasonbradley

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Jun 16, 2024

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Simplifying (-2a^3b^2c^0/3a^2b^3c^7)^-2

This problem involves simplifying an expression with exponents and fractions. Let's break it down step-by-step:

Understanding the Properties of Exponents

• Negative Exponent: A term raised to a negative exponent is equivalent to its reciprocal raised to the positive version of the exponent. For example, x^-2 = 1/x^2.
• Fractional Exponent: A term raised to a fractional exponent is equivalent to the root of the term. For example, x^(1/2) = √x.
• Product of Exponents: When multiplying exponents with the same base, add the powers. For example, x^m * x^n = x^(m+n).
• Division of Exponents: When dividing exponents with the same base, subtract the powers. For example, x^m / x^n = x^(m-n).
• Zero Exponent: Any non-zero number raised to the power of zero equals 1. For example, x^0 = 1.

Simplifying the Expression

1. Apply the Negative Exponent Rule: (-2a^3b^2c^0/3a^2b^3c^7)^-2 = (3a^2b^3c^7/-2a^3b^2c^0)^2

2. Simplify the Expression Inside the Parentheses: (3a^2b^3c^7/-2a^3b^2c^0)^2 = (3/(-2)) * (a^(2-3)) * (b^(3-2)) * (c^(7-0))^2

3. Simplify Further: (-3/2) * (a^-1) * (b^1) * (c^7)^2

4. Apply the Exponent Rule for a Product: (-3/2) * (a^-1) * (b^1) * (c^14)

5. Apply the Negative Exponent Rule: (-3/2) * (1/a) * (b) * (c^14)

6. Combine Terms: (-3b*c^14)/(2a)

Therefore, the simplified form of the expression (-2a^3b^2c^0/3a^2b^3c^7)^-2 is (-3b*c^14)/(2a).