Match each expression in Column I with its equivalent expression in Column II.

Verified step by step guidance

1. Step 1: Understand that expressions with fractional exponents like \(a^{m/n}\) can be rewritten as \(\left(\sqrt[n]{a}\right)^m\) or \(\sqrt[n]{a^m}\). This helps simplify the expressions.
2. Step 2: For each expression in Column I, rewrite the base and exponent to identify its simplified form. For example, \((4/9)^{3/2}\) can be seen as \(\left(\sqrt{4/9}\right)^3\).
3. Step 3: Calculate the square root of the base inside the parentheses. For \((4/9)\), \(\sqrt{4/9} = 2/3\). Then raise this result to the power of 3: \((2/3)^3\).
4. Step 4: For negative exponents like \((4/9)^{-3/2}\), recall that \(a^{-m} = \frac{1}{a^m}\). So rewrite the expression as the reciprocal of the positive exponent form.
5. Step 5: Pay attention to the negative signs outside the expressions in parts c and d, which will affect the final sign of the simplified value. After simplifying the magnitude, apply the negative sign accordingly.

Key Concepts

Rational Exponents

Rational exponents represent roots and powers simultaneously. For example, a^(m/n) means the nth root of a raised to the mth power. Understanding how to simplify expressions with fractional exponents is essential for matching equivalent forms.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For instance, a^(-m) = 1/(a^m). Recognizing this helps convert expressions with negative exponents into more familiar fractional forms.

Properties of Exponents and Simplification

Applying exponent rules, such as (a/b)^m = a^m / b^m, allows simplification of complex expressions. Combining these with sign considerations (positive or negative) is crucial to correctly match expressions with their equivalents.