Problem 23 - College Algebra with Modeling and Visualization

Question

Exercises \(17-34:\) Evaluate the expression by hand. Check your result with a calculator. $$ \left(\frac{2}{3}\right)^{3} $$

Step-by-Step Solution

Answer

The evaluated expression is \( \frac{8}{27} \).

Step 1: Understand the Expression

The exercise asks us to evaluate \( \left(\frac{2}{3}\right)^3 \). Here, \( \left(\frac{2}{3}\right)^3 \) means multiplying \( \frac{2}{3} \) by itself three times. This is equivalent to \( \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \).

Step 2: Multiply Fractions

First, multiply the numerators and then the denominators of the fractions. Start by multiplying the first two fractions: \( \frac{2}{3} \times \frac{2}{3} = \frac{2 \times 2}{3 \times 3} = \frac{4}{9} \).

Step 3: Multiply Result by the Third Fraction

Now, multiply the result from Step 2 by the third fraction: \( \frac{4}{9} \times \frac{2}{3} = \frac{4 \times 2}{9 \times 3} = \frac{8}{27} \).

Step 4: Verify with a Calculator

To ensure accuracy, calculate \( \left(\frac{2}{3}\right)^3 \) using a calculator. Input \( \frac{2}{3} \), raise it to the power of 3, and confirm the result is \( \frac{8}{27} \).

Key Concepts

Multiplying Fractions

When multiplying fractions, you start by dealing with two primary components: the numerators and the denominators. It's an easy process once you get familiar with it. Essentially, what you do is multiply the numerators together and then multiply the denominators together. For example, if you have the fractions \( \frac{2}{3} \times \frac{2}{3} \), you first multiply the numerators, which gives you \( 2 \times 2 = 4 \).
Next, multiply the denominators, resulting in \( 3 \times 3 = 9 \). Thus, \( \frac{2}{3} \times \frac{2}{3} \) equals \( \frac{4}{9} \). Take this one step further by multiplying the result with another fraction, for example, \( \frac{4}{9} \times \frac{2}{3} \), and you'll have \( \frac{4 \times 2}{9 \times 3} = \frac{8}{27} \). This intuitive method makes multiplying fractions a breeze!

Numerator and Denominator

Understanding the roles of the numerator and denominator is key when working with fractions. The numerator is the top number of the fraction, representing how many parts of a whole are being considered. Meanwhile, the denominator is the bottom number, indicating the number of equal parts that make up the whole.
In our earlier example \( \frac{2}{3} \), 2 is the numerator and 3 is the denominator. When multiplying fractions, keep in mind that each product's numerator comes from multiplying the numerators of the fractions being multiplied, while the product’s denominator comes from multiplying the denominators.
For clarity, always write each product above its corresponding denominator directly during the calculation. This will help you keep track of where each number goes, ensuring you end up with the correct fraction.

Calculator Verification

After calculating a fraction exponentiation manually, it's essential to verify your results with a calculator. This ensures accuracy and confidence in your answer. To check our example \( \left(\frac{2}{3}\right)^3 \) using a calculator, follow these steps:

• Input the fraction \( 2 \div 3 \).
• Use the exponentiation function (often labeled as \(^\wedge\) or \(x^y\)).
• Raise the fraction to the power of 3 by entering 3.
• Finally, examine the result, which should read \( \frac{8}{27} \) or its decimal equivalent.

By verifying calculations with a calculator, you not only double-check your work but also familiarize yourself with useful calculator functions, crucial for more complex problems.