Problem 73 - Algebra A Combined Function

Question

Evaluate each expression. $$ \left(\frac{7}{8}\right)^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{49}{64} \).

Step 1: Identify the Base and Exponent

The expression provided is \( \left(\frac{7}{8}\right)^2 \). Here, the base is the fraction \( \frac{7}{8} \), and the exponent is 2.

Step 2: Square the Numerator

To calculate \( \left(\frac{7}{8}\right)^2 \), start by squaring the numerator of the fraction. The numerator is 7, so compute \( 7^2 = 49 \).

Step 3: Square the Denominator

Next, square the denominator of the fraction. The denominator is 8, so compute \( 8^2 = 64 \).

Step 4: Formulate the Resulting Fraction

Combine the results from Steps 2 and 3 to form the fraction raised to the power of 2. The resulting fraction is \( \frac{49}{64} \).

Key Concepts

Understanding Base and Exponent in Fraction Exponentiation

When you encounter an expression like \( \left( \frac{7}{8} \right)^2 \), it's crucial to identify two main components: the base and the exponent. The base is the number or expression that is being multiplied by itself, and the exponent indicates how many times the base is multiplied. In this case, the base is the fraction \( \frac{7}{8} \), and the exponent is 2. This means the fraction needs to be multiplied by itself once more. Understanding these components is essential because it sets the stage for solving problems involving fraction exponentiation. Breaking it down:

• The base is \( \frac{7}{8} \), which is the fraction you will be working with.
• The exponent is 2, indicating the base will be squared or multiplied by itself.

This understanding helps you move confidently to the next steps of fraction exponentiation.

Numerator and Denominator in Fraction Operations

Fractions consist of two parts: the numerator and the denominator. When performing operations on fractions, it's important to recognize the role each part plays. The numerator is the top number and represents the quantity of parts being considered, while the denominator, the bottom number, indicates the total number of parts that make up a whole. Using the fraction \( \frac{7}{8} \), here:

• The numerator is 7, meaning we have seven parts.
• The denominator is 8, showing each whole is divided into eight parts.

During exponentiation, both parts must be squared separately. First, you square the numerator: \( 7^2 = 49 \). Then, you square the denominator: \( 8^2 = 64 \). Keeping track of these parts separately ensures accurate and efficient results when dealing with fractional exponentiation.

Squaring Fractions: How It Works

To square a fraction means to multiply the fraction by itself. This involves squaring both the numerator and the denominator, simplifying the process by breaking it into manageable parts. Let's explore squaring the fraction \( \frac{7}{8} \):

• First, you square the numerator: \( 7 \times 7 = 49 \).
• Then, you square the denominator: \( 8 \times 8 = 64 \).

After these calculations, the squared fraction is \( \frac{49}{64} \). The process of squaring fractions highlights the beauty of mathematics, where complex operations become simple through structured approaches. Always remember to perform operations separately on the numerator and denominator, then combine your results for the final answer.