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: Apply the zero exponent rule to simplify \((9x^3y^{-3})^0\) to 1, since any nonzero number raised to the power of 0 is 1.
2. Step 2: Simplify each term with negative exponents by applying the rule \((a^m)^n = a^{m \cdot n}\). For example, \((2^{-1}x^{-3}y^{-1})^{-2}\) becomes \(2^2x^6y^2\).
3. Step 3: Simplify \((2x^{-6}y^4)^{-2}\) by applying the same rule, resulting in \(2^{-2}x^{12}y^{-8}\).
4. Step 4: Simplify the denominator \((2x^{-4}y^{-6})^2\) by applying the power of a power rule, resulting in \(2^2x^{-8}y^{-12}\).
5. Step 5: Combine all the simplified expressions and apply the quotient rule \(\frac{a^m}{a^n} = a^{m-n}\) to simplify the entire expression.

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 base 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 have a zero exponent, as it can significantly reduce the complexity of the expression.