Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions.$$ \frac{14}{21}\left(\frac{2}{5-\frac{1}{3}}\right)^{2} $$

Step-by-Step Solution

Answer

\(\frac{6}{49}\)

Step 1: Simplify the Inner Expression

First, let's simplify the expression inside the parentheses: \[5 - \frac{1}{3}\]To do this, convert 5 to a fraction with a denominator of 3 to subtract the fractions:\[5 = \frac{15}{3}\]Thus, we have:\[\frac{15}{3} - \frac{1}{3} = \frac{14}{3}\]So, the expression becomes:\[\left(\frac{2}{\frac{14}{3}}\right)^2\]This is equivalent to:\[\left(\frac{2 \times 3}{14}\right)^2 = \left(\frac{6}{14}\right)^2\]

Step 2: Simplify the Fraction Inside the Square

Now, simplify the fraction \(\frac{6}{14}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:\[\frac{6}{14} = \frac{3}{7}\]Thus, the expression inside the square becomes:\[\left(\frac{3}{7}\right)^2\]

Step 3: Calculate the Square

Next, calculate the square of \(\frac{3}{7}\):\[\left(\frac{3}{7}\right)^2 = \frac{3^2}{7^2} = \frac{9}{49}\]This means the expression simplifies to:\[\frac{14}{21} \times \frac{9}{49}\]

Step 4: Simplify the Coefficient

Simplify the coefficient \(\frac{14}{21}\) by dividing the numerator and the denominator by their greatest common divisor, which is 7:\[\frac{14}{21} = \frac{2}{3}\]Multiply this by the simplified fraction from the previous step:\[\frac{2}{3} \times \frac{9}{49} = \frac{2 \times 9}{3 \times 49} = \frac{18}{147}\]

Step 5: Reduce the Resulting Fraction

Finally, simplify \(\frac{18}{147}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3:\[\frac{18}{147} = \frac{6}{49}\]The expression is simplified to the fraction \(\frac{6}{49}\).

Key Concepts

Fraction Simplification

Fraction simplification is a crucial skill when working with algebraic expressions. It involves reducing a fraction to its simplest form, making it easier to work with in calculations.
Fraction simplification is often achieved by dividing both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). The GCD is the largest number that evenly divides both the numerator and the denominator.
Let's take the fraction \(\frac{14}{21}\) as an example. The GCD of 14 and 21 is 7 since it's the largest number that both 14 and 21 can be divided by without a remainder.

• Divide the numerator by 7: \(14 \div 7 = 2\)
• Divide the denominator by 7: \(21 \div 7 = 3\)

So, \(\frac{14}{21}\) simplified becomes \(\frac{2}{3}\). It's essential to simplify fractions to make arithmetic operations more straightforward and ensure accurate results.

Arithmetic Operations

Arithmetic operations refer to basic mathematical calculations including addition, subtraction, multiplication, and division. In the context of algebraic expressions, you use these operations to simplify expressions to their simplest forms.
When dealing with fractions in algebraic expressions, understanding arithmetic operations becomes crucial. For instance, simplifying \(5 - \frac{1}{3}\) involves converting 5 into a fraction so both terms have a common denominator, making them easier to subtract.

• Convert 5 to \(\frac{15}{3}\) since \(5 \times 3 = 15\).
• Subtract: \(\frac{15}{3} - \frac{1}{3} = \frac{14}{3}\).

After subtraction, operations also include managing powers and multiplication. For \( \left(\frac{2}{\frac{14}{3}}\right)^2\), you invert the denominator to multiply: \(\frac{2 \times 3}{14} = \frac{6}{14}\). Finally, simplify by division to reach its simplest form. Mastery of these operations is key to effectively tackling similar algebraic expressions.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is fundamental in simplifying fractions. It is defined as the largest number that evenly divides two or more numbers without leaving a remainder.
to identify the gcd of two numbers, list the factors of each number and recognize the largest common factor. Consider the fraction \(\frac{6}{14}\), where the GCD of 6 and 14 must be found.

• Factors of 6: 1, 2, 3, 6
• Factors of 14: 1, 2, 7, 14

The largest number they share is 2, so the GCD is 2. By dividing both the numerator and the denominator by the GCD, we simplify \(\frac{6}{14}\) to \(\frac{3}{7}\).
Using the GCD not only helps in reducing fractions but also in simplifying complex algebraic expressions, ensuring calculations are more manageable without altering the value of the fraction. It maximizes simplification, an essential step in algebra.