Problem 30 - Brief Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{125}{8}\right)^{2 / 3} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{25}{4} \).

Step 1: Understand the Expression

The given expression is \( \left(\frac{125}{8}\right)^{2/3} \). The fraction \( \frac{2}{3} \) in the exponent indicates that we need to perform both a cube root and a square operation. Specifically, \( x^{2/3} \) can be interpreted as \( \left( x^{1/3} \right)^2 \). Therefore, we will first find the cube root of \( \frac{125}{8} \), and then square the result.

Step 2: Calculate the Cube Root

To find the cube root of \( \frac{125}{8} \), we can calculate \( \sqrt[3]{125} \) and \( \sqrt[3]{8} \) separately.\\( \sqrt[3]{125} = 5 \) because \( 5^3 = 125 \).\\( \sqrt[3]{8} = 2 \) because \( 2^3 = 8 \).\Therefore, \( \sqrt[3]{\frac{125}{8}} = \frac{5}{2} \).

Step 3: Square the Result

After finding the cube root, we now square \( \frac{5}{2} \).\\( \left(\frac{5}{2}\right)^2 = \frac{5^2}{2^2} = \frac{25}{4} \).

Step 4: Simplify the Expression (if needed)

The expression \( \frac{25}{4} \) is already in its simplest form as a fraction. There is no need for further simplification.

Key Concepts

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number back. To find the cube root of a fraction like \( \frac{125}{8} \), you can approach by finding the cube root of both the numerator and the denominator separately.
For \( 125 \):

• Calculate \( \sqrt[3]{125} \) as 5 because \( 5^3 = 125 \).

For \( 8 \):

• Calculate \( \sqrt[3]{8} \) as 2 because \( 2^3 = 8 \).

Thus, \( \sqrt[3]{\frac{125}{8}} = \frac{5}{2} \). This intermediary step is crucial before performing additional operations like squaring.

Fractional Exponents

Fractional exponents can seem a little tricky at first, but they are quite manageable once you understand their meaning. An exponent in the form of a fraction, such as \( \frac{2}{3} \), tells you to perform two operations:
1. The denominator indicates the root to be taken, in this case, the cube root.
2. The numerator indicates the power to which the result must be raised, in this case squared.
For the expression \( \left(x^{2/3}\right) \), you first find \( x^{1/3} \), or the cube root, and then square it to get \( \left(x^{1/3}\right)^2 \). This systematic method ensures clarity and precision in evaluation.

Simplifying Fractions

Simplification is the process of reducing a fraction to its simplest form, making it easier to understand and work with. After determining cube roots and performing the subsequent steps in our problem, we arrive at \( \frac{25}{4} \).
Simplify by checking if the numerator and denominator have any common factors. In \( \frac{25}{4} \), they don't have any, as 25 is only divisible by 1 and 25, and 4 is only divisible by 1 and 2. Thus, it's already as simple as possible. This verification step helps prevent potential errors and confusion.

Step-by-step Solutions

Breaking down a complex problem into manageable steps helps build confidence and understanding. The solution for \( \left(\frac{125}{8}\right)^{2/3} \) is approached systematically:

• **Step 1:** Identify the expression, recognizing the role of fractional exponents.
• **Step 2:** Calculate the cube root of the fraction to simplify the base.
• **Step 3:** Square the result of the cube root operation.
• **Step 4:** Simplify the result, if possible, to get the final answer.

This method not only provides the correct answer but also reinforces the underlying principles of the mathematical concepts involved, leading to a deeper understanding.