Problem 50 - Prealgebra

Question

Expand and simplify each of the following. $$\left(\frac{5}{6}\right)^{2} \cdot \frac{12}{15}$$

Step-by-Step Solution

Answer

The expanded and simplified form of \( \left(\frac{5}{6}\right)^{2} \cdot \frac{12}{15} \) is \( \frac{5}{9} \).

Step 1: Evaluate the Power

Calculate \( \left(\frac{5}{6}\right)^{2} \) by squaring both the numerator and the denominator.\[ \left(\frac{5}{6}\right)^{2} = \frac{5^2}{6^2} = \frac{25}{36} \]

Step 2: Set Up the Multiplication

Write down the expression to multiply the result from Step 1 by \( \frac{12}{15} \).\[ \frac{25}{36} \times \frac{12}{15} \]

Step 3: Multiply the Numerators

Multiply the numerators of the fractions:\[ 25 \times 12 = 300 \]

Step 4: Multiply the Denominators

Multiply the denominators of the fractions:\[ 36 \times 15 = 540 \]

Step 5: Form the Resulting Fraction

Combine the products of the numerators and denominators to form a new fraction:\[ \frac{300}{540} \]

Step 6: Simplify the Fraction

Find the greatest common divisor (GCD) of 300 and 540, which is 60. Divide both numerator and denominator by this GCD:\[ \frac{300 \div 60}{540 \div 60} = \frac{5}{9} \]

Key Concepts

Simplifying Fractions

Simplifying fractions means reducing them to their smallest form, where the numerator (top number) and the denominator (bottom number) no longer have any common factors other than 1. For example, to simplify the fraction \( \frac{300}{540} \), as in our exercise, we look for a common divisor. By identifying the greatest common divisor (GCD), which in this case is 60, you divide both the top and bottom numbers by this GCD. Here are the steps:

• Identify the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.
• Rewrite the fraction with the new numbers for numerator and denominator.

In our solution, dividing 300 and 540 by 60 gives us \( \frac{5}{9} \), the simplified fraction. This process makes fractions much easier to work with and compares them more effectively.

Multiplying Fractions

Multiplying fractions is a simple process. It involves multiplying the numerators (top numbers) together to get a new numerator, and multiplying the denominators (bottom numbers) together to get a new denominator. In our exercise, once we had \( \frac{25}{36} \), we multiplied it by \( \frac{12}{15} \). Here's how you do it:

• Multiply the numerators: \( 25 \times 12 = 300 \).
• Multiply the denominators: \( 36 \times 15 = 540 \).
• Write down the fraction with the new numbers: \( \frac{300}{540} \).

After you have the multiplication result, you often need to simplify the fraction, as fractions can become quite large. In this exercise, simplifying further gives us \( \frac{5}{9} \), showing the difference between just multiplying and finding the simplest form.

Powers and Exponents

Powers and exponents are a way to express repeated multiplication of the same number. In the exercise, you see \( \left(\frac{5}{6}\right)^2 \) which means the fraction \( \frac{5}{6} \) is multiplied by itself. This process is crucial for understanding how to handle fractions with exponents:

• Raise both the numerator and the denominator to the power indicated by the exponent.
• For example, \( \left(\frac{5}{6}\right)^2 = \frac{5^2}{6^2} = \frac{25}{36} \).

This provides a simplified fraction that is easier to use in further operations like multiplication or division. By squaring both the top and bottom numbers separately, we maintain the balance of the fraction even if they are much larger or smaller. Understanding this concept helps facilitate work with more complex algebraic expressions involving fractions.