Problem 8 - 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) $$-\left(\frac{27}{8}\right)^{-2 / 3}$$

Step-by-Step Solution

Answer

Expression (I) matches with choice C in Group II: \(-\frac{4}{9}\).

Step 1: Simplify the Expression

Let's start by simplifying the expression given in Group I, which is \(-\left(\frac{27}{8}\right)^{-2/3}\). First, address the exponent \(-2/3\). According to the rules of exponents, a negative exponent \(a^{-n}\) becomes \(\frac{1}{a^n}\).

Step 2: Apply the Negative Exponent

Convert the negative exponent to a positive by rewriting the expression: \(-\left(\frac{27}{8}\right)^{-2/3} = -\frac{1}{\left(\frac{27}{8}\right)^{2/3}}\). This step removes the negative sign from the exponent.

Step 3: Compute the Fractional Exponent

Next, evaluate \(\left(\frac{27}{8}\right)^{2/3}\). The fraction \(2/3\) implies both a square and a cube root (specifically: take the cube root, then square the result). Begin by finding the cube root of each part: \(\sqrt[3]{27} = 3\) and \(\sqrt[3]{8} = 2\).

Step 4: Square the Result of the Cube Root

Now, square the results obtained from the cube roots: \(3^2 = 9\) and \(2^2 = 4\). This gives \(\left(\frac{27}{8}\right)^{2/3} = \frac{9}{4}\).

Step 5: Apply the Negative Sign

Return to the original problem, which included a negative sign outside the fraction: \(-\frac{1}{\left(\frac{27}{8}\right)^{2/3}} = -\frac{1}{\frac{9}{4}} = -\frac{4}{9}\).

Step 6: Match with Group II

Finally, compare \(-\frac{4}{9}\) to the options in Group II. It matches with option C, which is \(-\frac{4}{9}\).

Key Concepts

Exponents

Exponents are powerful tools in mathematics, used to simplify expressions and denote repeated multiplication of a number by itself. When you see an expression like \(x^n\), it means that \(x\), the base, is multiplied by itself \(n\) times. Here are some essential aspects to remember about exponents:

• **Zero Exponent**: Any non-zero number raised to the power of zero equals one, so \(a^0 = 1\).
• **Positive Exponents**: Indicate regular multiplication, such as \(a^3 = a \times a \times a\).
• **Power of a Product**: The rule \((ab)^n = a^n \times b^n\) allows the exponent to apply to each factor in the product.
• **Power of a Quotient**: Similar to products, the rule \((a/b)^n = a^n / b^n\) is used for division.

Understanding these rules is essential to manipulate and simplify algebraic expressions efficiently. They form the foundation upon which more complex operations, such as those involving fractional and negative exponents, are built.

Fractional Exponents

Fractional exponents represent both root and power operations within a single expression. For instance, the expression \(a^{m/n}\) can be interpreted as follows:

• **Roots and Powers Combined**: The fraction \(m/n\) implies that the base \(a\) should be raised to the power \(m\) after taking the \(n\)-th root.
• **Two Steps**: You first find the \(n\)-th root of \(a\), and then raise the result to the power of \(m\). For example, \(a^{2/3} = (\sqrt[3]{a})^2\).
• **Flexibility**: Depending on the expression, it may sometimes be more convenient to execute the power operation first, followed by the root.

These steps help break complex expressions into manageable pieces, making calculations easier. In our original problem, \(\left(\frac{27}{8}\right)^{2/3}\) required us to find the cube root of both the numerator and the denominator before squaring the results. This demonstrates how fractional exponents simplify the operation of both roots and powers simultaneously.

Negative Exponents

Negative exponents are a way of expressing reciprocals in the context of exponents. In simpler terms, \(a^{-n}\) is the same as \(1/a^n\). Here's how to think about negative exponents:

• **Reciprocal Rule**: The negative exponent indicates a reciprocal of the base raised to the corresponding positive exponent. Thus, \(2^{-3} = 1/2^3\).
• **Simplifying Expressions**: By converting negative exponents to positive, you can make calculations more straightforward. As shown in our exercise, \(-\left(\frac{27}{8}\right)^{-2/3} = -\frac{1}{\left(\frac{27}{8}\right)^{2/3}}\).
• **Expression Clarity**: This alteration helps clear up expressions that might initially look complex, replacing negative exponents with division, making manipulation easier.

These key ideas about negative exponents are necessary to solve algebraic expressions involving inverse operations concisely. It's important to remember that they serve the primary function of inversion while retaining the power operation of the positive counterpart.