Match each expression in Column I with its equivalent expression in Column II. See Example 8. a. (4/9)^3/2 b. (4/9)^-3/2 c. -(9/4)^3/2 d. -(9/4)^-3/2 A. 27/8 B. -27/8 C. 8/27 D. -8/27

Verified step by step guidance

1. Step 1: Understand the expressions and the exponents. Each expression involves a base raised to a fractional exponent, which can be interpreted as a root and a power. For example, \(a^{m/n} = \sqrt[n]{a^m}\).
2. Step 2: Simplify the bases where possible. Notice that \(\frac{4}{9}\) and \(\frac{9}{4}\) are reciprocals. This relationship will help when dealing with negative exponents, since \(a^{-x} = \frac{1}{a^x}\).
3. Step 3: For each expression, rewrite the fractional exponent as a root and a power. For example, \((\frac{4}{9})^{3/2} = \left(\sqrt{\frac{4}{9}}\right)^3\).
4. Step 4: Calculate the square root of the base fraction first, then raise the result to the power indicated. For example, \(\sqrt{\frac{4}{9}} = \frac{2}{3}\), so \((\frac{4}{9})^{3/2} = \left(\frac{2}{3}\right)^3\).
5. Step 5: Apply the negative sign and negative exponents carefully. For negative exponents, take the reciprocal of the base raised to the positive exponent. For example, \((\frac{4}{9})^{-3/2} = \frac{1}{(\frac{4}{9})^{3/2}}\).

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, or (√[n]{a})^m. Understanding how to manipulate these helps simplify and compare expressions involving fractional powers.

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 allows you to rewrite expressions with negative exponents into more manageable forms.

Properties of Exponents and Simplification

Using exponent rules such as (a/b)^m = a^m / b^m and distributing powers over multiplication or division is essential. Simplifying expressions by applying these rules helps match equivalent forms and evaluate numerical values accurately.