In Exercises 117–124, simplify each exponential expression.9y⁴/x⁻² + (x⁻¹/y²)⁻²

Verified step by step guidance

1. Identify the components of the expression: \( \frac{9y^4}{x^{-2}} + \left(\frac{x^{-1}}{y^2}\right)^{-2} \).
2. Simplify the first term \( \frac{9y^4}{x^{-2}} \) by using the property of exponents \( a^{-n} = \frac{1}{a^n} \), which gives \( 9y^4 \cdot x^2 \).
3. Simplify the second term \( \left(\frac{x^{-1}}{y^2}\right)^{-2} \) by applying the power of a quotient rule \( \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n \), resulting in \( \left(\frac{y^2}{x^{-1}}\right)^2 \).
4. Further simplify \( \left(\frac{y^2}{x^{-1}}\right)^2 \) by rewriting \( x^{-1} \) as \( \frac{1}{x} \), leading to \( \left(y^2 \cdot x\right)^2 \).
5. Combine the simplified expressions: \( 9y^4x^2 + y^4x^2 \).

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 quotient of powers (a^m / a^n = a^(m-n)), and the power of a power ( (a^m)^n = a^(m*n)). Understanding these rules is essential for simplifying expressions with exponents effectively.

Negative Exponents

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

Reciprocal and Power of a Fraction

The reciprocal of a fraction is obtained by flipping the numerator and denominator, which is particularly relevant when dealing with negative exponents. Additionally, when raising a fraction to a power, both the numerator and denominator are raised to that power. This understanding is vital for correctly simplifying expressions that involve fractions and exponents.