Problem 5 - 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) $$\left(\frac{8}{27}\right)^{2 / 3}$$

Step-by-Step Solution

Answer

The expression \( \left( \frac{8}{27} \right)^{2/3} \) matches with option D: \( \frac{4}{9} \).

Step 1: Understand the Expression

Identify the task, which is to evaluate the expression \( \left( \frac{8}{27} \right)^{2/3} \) and match it with the options A to H. The expression is in fractional form raised to a rational exponent.

Step 2: Rewrite the Expression

Rewrite the expression \( \left( \frac{8}{27} \right)^{2/3} \) in terms of its components. This means taking both the numerator and the denominator of the fraction to the power of \(2/3\): \[\left( \frac{8}{27} \right)^{2/3} = \frac{8^{2/3}}{27^{2/3}}\]

Step 3: Simplify the Fractional Exponent

To simplify \( \frac{8^{2/3}}{27^{2/3}} \), first calculate the cube roots (the denominator of the exponent is 3) of each number, then square the results (the numerator of the exponent is 2): \[8^{2/3} = (8^{1/3})^2 = (2)^2 = 4\]\[27^{2/3} = (27^{1/3})^2 = (3)^2 = 9\]

Step 4: Evaluate the Expression

Using the results from Step 3, substitute back into the expression:\[\frac{8^{2/3}}{27^{2/3}} = \frac{4}{9}\]Which matches choice D.

Key Concepts

Fractional Exponents

Fractional exponents, also known as rational exponents, are a way to express roots and powers in a single operation. They give you an alternative to using radical signs, which often makes calculations simpler and more straightforward in algebra. The numerator of the fractional exponent () indicates the power and the denominator () indicates the root.
For example, the expression \((8^{2/3})\) involves both a power and a root:

• The denominator 3 indicates a cube root.
• The numerator 2 indicates that the cube root should be squared.

Thus, \((8^{2/3})\) tells us to take the cube root of 8 and then square the result. This ability to break down the process into smaller steps helps in understanding and simplifying complex expressions.

Simplification Techniques

Simplification techniques are crucial when dealing with fractional exponents. Breaking down expressions into more manageable steps can help you tackle them with ease. A common method is to simplify the roots before applying the exponent, particularly when using rational exponents.
In the provided exercise, the expression \[ \left( \frac{8}{27} \right)^{2/3} \] is simplified by addressing each part of the fraction separately. Here's the breakdown:

• Find the cube root of the numerator (8), which equals 2, and then square it to get 4.
• Similarly, find the cube root of the denominator (27), which equals 3, and then square it to get 9.

Combining these steps simplifies the original expression to \(( \frac{4}{9} )\). By breaking it down like this, you follow a more structured approach, making it easier to handle each component of the expression.

Exponentiation

Exponentiation is a fundamental mathematical operation involving numbers, expressing repeated multiplication. When a number (base) is raised to an exponent, it means multiplying the base by itself as many times as indicated by the exponent.
With rational exponents, you combine the concepts of roots and powers. For instance, evaluating \((8^{2/3})\) involves understanding exponentiation at two levels:

• The fractional exponent \(((2/3))\) tells you two operations: first take the cube root, then square.
• Because finding cube roots and squaring are essentially repeated multiplications, these steps adhere to the same laws of exponents applied differently.

Having a strong grasp of basic exponentiation rules will help you decode fractional exponents effectively, as you apply the principles of multiplying bases, dividing in roots, and raising to additional powers.