Problem 4 - A Graphical Approach to College Algebra

Question

Concept Check Match each expression from Group I with the correct choice from Group II. Choices may be used once, more than once, or not at all. (II) \(\begin{array}{lll}\text { A. } \frac{9}{4} & \text { B. }-\frac{9}{4}\end{array}\) \(\begin{array}{lll}\text { C. }-\frac{4}{9} & \text { D. } \frac{4}{9}\end{array}\) E. \(\frac{8}{27} \quad\) F. \(-\frac{27}{8}\) G. \(\frac{27}{8} \quad\) H. \(-\frac{8}{27}\) (I) $$-\left(\frac{4}{9}\right)^{-3 / 2}$$

Step-by-Step Solution

Answer

The expression matches with choice F, \(-\frac{27}{8}\).

Step 1: Understand the problem

We are provided with an expression in Group I that we need to evaluate and match with one of the choices in Group II. The given expression to evaluate is \(-\left(\frac{4}{9}\right)^{-3/2}\).

Step 2: Simplify the expression

The expression \(\left(\frac{4}{9}\right)^{-3/2}\) involves a fractional exponent. A negative exponent indicates a reciprocal, and raising to the \(3/2\) power involves both a square root and a cube.

Step 3: Use the negative exponent property

Apply the negative exponent rule: \(\left(a^{-n}\right) = \frac{1}{a^n}\). Therefore, \(\left(\frac{4}{9}\right)^{-3/2} = \frac{1}{\left(\frac{4}{9}\right)^{3/2}}\).

Step 4: Evaluate \(\left(\frac{4}{9}\right)^{3/2}\)

First, compute the square root: \(\left(\frac{4}{9}\right)^{1/2}= \frac{2}{3}\). Then, raise this result to the power of 3: \(\left(\frac{2}{3}\right)^3 = \frac{8}{27}\).

Step 5: Calculate the reciprocal

Now, take the reciprocal from Step 3: \(\frac{1}{\frac{8}{27}} = \frac{27}{8}\).

Step 6: Determine the sign

Finally, incorporate the initial negative sign in the expression: \(-\left(\frac{27}{8}\right) = -\frac{27}{8}\).

Step 7: Match with Group II

The calculated value \(-\frac{27}{8}\) corresponds to choice F in Group II.

Key Concepts

Negative Exponent

Let's begin with understanding the concept of a negative exponent. A negative exponent essentially means we are dealing with the reciprocal of a base raised to a positive exponent. For instance, if you have an expression like \(a^{-n}\), it can be rewritten as \(\frac{1}{a^n}\). This means, instead of multiplying the base by itself \(n\) times, you're dividing 1 by the base raised to the positive \(n\).

Why does this happen? The fundamental reason behind this is the definition of exponentiation. A negative exponent "undoes" or inverts the multiplication suggested by a positive exponent. Hence, when faced with an expression like \(\left(\frac{4}{9}\right)^{-3/2}\), you should immediately think of flipping the fraction and simplifying further. Remember:

• A negative exponent flips the base to its reciprocal.
• Then, you raise it to the corresponding positive exponent.

Reciprocal

Every fraction has a reciprocal. The reciprocal is formed by swapping the numerator and the denominator. If your original fraction is \(\frac{4}{9}\), the reciprocal is \(\frac{9}{4}\).

Reciprocals are especially helpful in operations involving division and negative exponents. For a negative exponent expression like \(\left(\frac{4}{9}\right)^{-3/2}\), computing its reciprocal is the first step to simplify it, which is found using the negative exponent rule. By flipping the fraction, it becomes easier to apply further operations.

Some key points to remember about reciprocals:

• The product of a number and its reciprocal is always 1. For example, \(\frac{4}{9} \times \frac{9}{4} = 1\).
• Reciprocals lead to simplification in fractional exponent expressions.

Power of a Fraction

Understanding how to handle powers of a fraction is fundamental when working with fractional exponents. The power indicates how many times you multiply the fraction by itself. However, a fractional exponent like \(\left(\frac{4}{9}\right)^{3/2}\) involves both the cube (3) and the square root (\(1/2\)).

Here's how you can tackle a power of a fraction step by step:

Firstly, you need to separate the operations:

• Begin by taking the square root of the fraction, which simplifies \(\left(\frac{4}{9}\right)^{1/2}\) to \(\frac{2}{3}\).
• Next, raise the result to the power of 3. So, \(\left(\frac{2}{3}\right)^3\) becomes \(\frac{8}{27}\).

This dual operation of roots and powers exemplifies why careful handling of order is crucial. Completing both steps ensures accurate simplification in problems involving fractional exponents.