Problem 7 - 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

Match with choice B: \(-\frac{9}{4}\).

Step 1: Understand the Expression

Our task is to simplify and evaluate the expression given in Group I, which is \(-\left(\frac{27}{8}\right)^{2/3}\). This means we need to take \(\frac{27}{8}\), raise it to the power of \(\frac{2}{3}\), and then apply a negative sign.

Step 2: Simplify the Base

The base of our expression is \(\frac{27}{8}\). First, realize that \(27 = 3^3\) and \(8 = 2^3\). Therefore, we can rewrite the base as \(\left(\frac{3}{2}\right)^3\).

Step 3: Apply the Exponent

Next, we raise \(\left(\frac{3}{2}\right)^3\) to the \(\frac{2}{3}\) power, which can be written as:\[\left(\left(\frac{3}{2}\right)^3\right)^{2/3} = \left(\frac{3}{2}\right)^{3 \times \frac{2}{3}} = \left(\frac{3}{2}\right)^2\]

Step 4: Calculate \(\left(\frac{3}{2}\right)^2\)

Now, compute \(\left(\frac{3}{2}\right)^2\):\[\left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4}\]

Step 5: Apply the Negative Sign

The last step is to apply the negative sign to the result of the previous step:\[-\left(\frac{9}{4}\right) = -\frac{9}{4}\]

Step 6: Match with Group II Choices

Compare \(-\frac{9}{4}\) to the options in Group II. The correct match is option B.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Unlike regular numbers, these expressions can include variables, but in this exercise, they only consist of numbers. Simplifying rational expressions involves breaking down the expression into its simplest form. This can often mean factoring the numerator and denominator, identifying common factors, and reducing

• The fraction form of these expressions allows for operations such as simplification, addition, and multiplication.
• Keep in mind, the denominator should not be zero, as division by zero is undefined.

Understanding rational expressions is crucial when dealing with algebraic equations and their simplification. Let's look at how these expressions can relate to exponents.

Exponents

Exponents are used to express repeated multiplication of the same number. In algebra, they provide a convenient way to work with large numbers. When you see a number like \((3^4)\), it means \((3 \times 3 \times 3 \times 3)\). Exponents have specific rules that help you simplify complex mathematical expressions and make calculations more manageable.

• When multiplying numbers with the same base, you add their exponents, \(a^m \times a^n = a^{m+n}\).
• When dividing, you subtract: \(a^m \/ a^n = a^{m-n}\).
• Raising an exponent to another power multiplies them: \((a^m)^n = a^{m*n}\).

Recognizing how exponents work applies not only to individual numbers but also to entire expressions, as seen in our original example. This leads us directly into the concept of simplification.

Simplification

Simplification involves reducing an expression to its most basic or concise form without changing its value. The overall goal is to make the expression easier to work with by applying basic algebraic rules and operations.

• Simplification of numbers or expressions often involves factoring, canceling, and applying exponent rules.
• It's important to perform operations in the correct sequence; for example, in the expression given in the exercise, we first raise the fraction to the power, then apply the negative sign.

In the solved exercise, identifying \(27/8\) as \((3/2)^3\) allowed us to handle it effectively using the exponent \((2/3)\), demonstrating a core simplification technique. Let's delve into how fractional powers operate on these expressions.

Fractional Powers

Fractional powers involve exponents that are fractions. These exponents provide a bridge between roots and powers. When you encounter an expression like \(x^{\frac{m}{n}}\), it refers to taking the nth root of x raised to the mth power.

• The denominator of the fraction indicates the root, while the numerator shows which power to raise the base to after root extraction.
• For instance, in \((a^{\frac{m}{n}}=\sqrt[n]{a^m}\)), you can handle fractional powers by first computing the root and then the power, or vice versa.

This dual process provides versatility in arithmetic and algebra alike, making complex expressions more manageable. In the solution provided, we applied this concept to raise \(\frac{27}{8}\) to the power of \(\frac{2}{3}\), simplifying it step-by-step.