Problem 74 - Algebra A Combined Function

Question

Evaluate each expression. $$ \left(\frac{2}{3}\right)^{3} $$

Step-by-Step Solution

Answer

\( \frac{8}{27} \)

Step 1: Understand the Expression

The expression provided is \( \left(\frac{2}{3}\right)^{3} \). This is a power expression where \( \frac{2}{3} \) is the base, and 3 is the exponent.

Step 2: Evaluate the Exponentiation

To evaluate \( \left(\frac{2}{3}\right)^{3} \), you multiply \( \frac{2}{3} \) by itself three times: \[ \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) \]

Step 3: Multiply the Numerators

Multiply all the numerators together: \( 2 \times 2 \times 2 = 8 \). The product of the numerators is 8.

Step 4: Multiply the Denominators

Multiply all the denominators together: \( 3 \times 3 \times 3 = 27 \). The product of the denominators is 27.

Step 5: Compose the Resulting Fraction

Combine the results from steps 3 and 4 to form the fraction: \( \frac{8}{27} \).

Key Concepts

Fraction Multiplication

When it comes to multiplying fractions, the key is to remember that you multiply the numerators together and the denominators together. This separates fraction multiplication from adding or subtracting fractions, where the denominators must be the same. Each fraction consists of a top number, called the numerator, and a bottom number, known as the denominator. When you multiply fractions like \( \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \), follow these steps:

• Multiply all the numerators: \( 2 \times 2 \times 2 = 8 \)
• Multiply all the denominators: \( 3 \times 3 \times 3 = 27 \)
• Combine the products to create a new fraction: \( \frac{8}{27} \)

This process shows how simple yet effective fraction multiplication can be!

Numerator and Denominator

Every fraction is composed of two essential parts: the numerator and the denominator. Understanding these components is crucial.

• Numerator: This is the top number of a fraction and represents how many parts of the whole are being considered. In the fraction \( \frac{2}{3} \), 2 is the numerator, indicating two parts of something divided into three.
• Denominator: This is the bottom number, which shows into how many parts the whole is divided. In \( \frac{2}{3} \), 3 is the denominator, meaning the whole is split into three equal parts.

The terms might sound complicated at first, but they're straightforward once you get the hang of it! When you multiply fractions, each of these parts is multiplied separately.

Power Expressions

Power expressions, or exponentiation, simplify repeated multiplication. In an expression like \( \left(\frac{2}{3}\right)^3 \), you have a base, which is the fraction \( \frac{2}{3} \), and an exponent, in this case, 3. This means you multiply the base by itself as many times as the exponent indicates.For \( \left(\frac{2}{3}\right)^3 \), this involves these steps:

• Take the base \( \frac{2}{3} \)
• Multiply it three times: \( \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \)
• Simplify the process using fraction multiplication, as explained earlier.

Exponentiation, especially with fractions, might seem daunting, but following the orderly steps makes it manageable and clear!