In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers. (2^−1x^−3y^−1)^−2(2x^−6y^4)^−2(9x^3y^−3)^0/(2x^−4y^−6)^2

Verified step by step guidance

1. Step 1: Simplify each term in the numerator and denominator separately. Start with the first term \((2^{-1}x^{-3}y^{-1})^{-2}\). Apply the power rule \((a^m)^n = a^{m \cdot n}\) to distribute the \(-2\) exponent to each base: \(2^{(-1)(-2)}x^{(-3)(-2)}y^{(-1)(-2)} = 2^2x^6y^2\).
2. Step 2: Simplify the second term \((2x^{-6}y^4)^{-2}\). Again, apply the power rule \((a^m)^n = a^{m \cdot n}\): \(2^{1 \cdot -2}x^{-6 \cdot -2}y^{4 \cdot -2} = 2^{-2}x^{12}y^{-8}\).
3. Step 3: Simplify the third term \((9x^3y^{-3})^0\). Any expression raised to the power of 0 is equal to 1, so \((9x^3y^{-3})^0 = 1\).
4. Step 4: Simplify the denominator \((2x^{-4}y^{-6})^2\). Apply the power rule \((a^m)^n = a^{m \cdot n}\): \(2^{1 \cdot 2}x^{-4 \cdot 2}y^{-6 \cdot 2} = 2^2x^{-8}y^{-12}\).
5. Step 5: Combine all the simplified terms into a single expression. The numerator becomes \(2^2x^6y^2 \cdot 2^{-2}x^{12}y^{-8} \cdot 1\), and the denominator is \(2^2x^{-8}y^{-12}\). Use the product rule \(a^m \cdot a^n = a^{m+n}\) to combine like bases in the numerator, and then simplify the fraction by subtracting exponents for like bases \(a^m / a^n = a^{m-n}\).

Key Concepts

Exponential Rules

Exponential rules are fundamental properties that govern the manipulation of expressions involving exponents. Key rules include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the power of a product ( (ab)^n = a^n * b^n). Understanding these rules is essential for simplifying complex exponential expressions.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, a^(-n) = 1/(a^n). This concept is crucial when simplifying expressions, as it allows for the transformation of negative exponents into a more manageable form, facilitating further simplification.

Zero Exponent Rule

The zero exponent rule states that any nonzero number raised to the power of zero equals one, expressed as a^0 = 1. This rule is important in simplifying expressions, particularly when dealing with terms that may have a zero exponent, as it can significantly reduce the complexity of the expression.