Problem 37 - Prealgebra

Question

Simplify each expression as much as possible. $$\frac{18}{35} \div\left(\frac{6}{7}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{7}{5} \).

Step 1: Understand the Expression

The expression we need to simplify is \( \frac{18}{35} \div \left( \frac{6}{7} \right)^2 \). It involves division of a fraction by the square of another fraction.

Step 2: Square the Fraction

First, we square the fraction \( \frac{6}{7} \). To do this, square both the numerator and the denominator separately: \( \left( \frac{6}{7} \right)^2 = \frac{6^2}{7^2} = \frac{36}{49} \).

Step 3: Rewrite the Division as Multiplication

Instead of dividing by a fraction, we multiply by its reciprocal. Therefore, \( \frac{18}{35} \div \frac{36}{49} \) can be rewritten as \( \frac{18}{35} \times \frac{49}{36} \).

Step 4: Simplify the Multiplication

Multiply the numerators and the denominators: \( \frac{18}{35} \times \frac{49}{36} = \frac{18 \times 49}{35 \times 36} \).

Step 5: Cancel Common Factors

Look for common factors between the numerators and denominators:- The factors of 18 are 2, 3, 3.- The factors of 35 are 5, 7.- The factors of 49 are 7, 7.- The factors of 36 are 2, 2, 3, 3. Cancel out the common factor of 7 in the numerator and denominator, and simplify:\[ \frac{18 \cdot 49}{35 \cdot 36} = \frac{18}{5} \cdot \frac{7}{18} = \frac{7}{5} \].

Step 6: Final Simplified Expression

The simplified version of the given expression is \( \frac{7}{5} \).

Key Concepts

Squaring Fractions

Squaring a fraction might sound complicated, but it is, in fact, quite straightforward. To square a fraction, you simply square both the numerator and the denominator separately. Consider the fraction \( \frac{6}{7} \). When we square it, we multiply the numerator and the denominator by themselves.
So it becomes:

• Numerator: \( 6^2 = 36 \)
• Denominator: \( 7^2 = 49 \)

This results in \( \left( \frac{6}{7} \right)^2 = \frac{36}{49} \). Remember, squaring maintains the fractional relationship as it involves both the top and bottom parts of the fraction. By applying this operation, you'll find it helpful in simplifying more complex expressions involving fractions.

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is key when dealing with division operations involving fractions. A reciprocal flips the numerator and the denominator of a fraction.
For instance, the reciprocal of \( \frac{36}{49} \) is \( \frac{49}{36} \). This concept is crucial because dividing by a fraction is the same as multiplying by its reciprocal.

• If you have \( a \div b \), you can rewrite it as \( a \times \frac{1}{b} \).

This transformation is what allows multiplication instead of division and simplifies calculations significantly. By using reciprocals, expressions become easier to handle, particularly when simplifying complex fractions.

Simplifying Expressions

Simplifying expressions is about reducing them to their simplest form. When you have an expression like \( \frac{18}{35} \div \left( \frac{6}{7} \right)^2 \), you start by converting the division into multiplication through the use of reciprocals.
The expression changes to \( \frac{18}{35} \times \frac{49}{36} \). Now, focus on simplifying the multiplication by finding the greatest common factors between numerators and denominators. By rewriting the division as multiplication, you streamline the process of simplification.
Simplifying expressions often involves several steps, including multiplying the fractions together and then cancelling out common factors. This approach helps you manage and solve math problems with more efficiency.

Common Factors

Common factors play a critical role in simplifying mathematical expressions. They refer to numbers that can divide two or more numbers without leaving a remainder.

• For instance, if you look at the expression \( \frac{18 \times 49}{35 \times 36} \), analyze each part for shared factors.
• The factors of 18 are 2, 3, and 3.
• The factors of 35 are 5 and 7.
• The factors of 49 are 7 and 7.
• The factors of 36 are 2, 2, 3, and 3.

Identifying common factors between numerators and denominators allows you to cancel them out and simplify the fraction. Here, both 7 and 18 appear in the multiplication, allowing you to reduce the fraction accordingly, ultimately resulting in \( \frac{7}{5} \). Paying attention to common factors is crucial for thorough and accurate simplification.