Problem 74 - Algebra A Combined Function

Question

Perform each indicated operation and simplify. $$ \frac{9}{2}\left(\frac{2}{9} x\right) $$

Step-by-Step Solution

Answer

The simplified expression is just \( x \).

Step 1: Distribute the Fraction

To begin, distribute the fraction \( \frac{9}{2} \) across the expression inside the parentheses, which is \( \frac{2}{9} x \). This means you multiply \( \frac{9}{2} \) by \( \frac{2}{9} x \).

Step 2: Multiply the Fractions

Multiply the fractions directly: \( \frac{9}{2} \times \frac{2}{9} = \frac{9 \times 2}{2 \times 9} = \frac{18}{18} \). This operation simplifies to 1.

Step 3: Simplify the Result

Since the result of multiplying the fractions is 1, the entire expression simplifies to \( 1 \times x = x \).

Key Concepts

Fraction Multiplication

When dealing with fractions in algebraic operations, fraction multiplication is a fundamental skill. You need to multiply the numerators (the numbers at the top of the fractions) and the denominators (the numbers at the bottom) separately.
For instance, in the example problem, we're tasked with multiplying the fractions \( \frac{9}{2} \) and \( \frac{2}{9} \). This is straightforward if you remember that:

• Multiply the numerators: \( 9 \times 2 = 18 \)
• Multiply the denominators: \( 2 \times 9 = 18 \)

After you get the results of these multiplications, you put them together to form a new fraction: \( \frac{18}{18} \).
This is a simple but crucial step, paving the way towards simplification.

Simplification

Simplification is all about reducing an expression to its simplest form. In the case of fractions, this means finding the smallest number that can divide both the numerator and the denominator evenly and then dividing both by this number.
After multiplying the fractions in our example (\( \frac{18}{18} \)), you will notice that both the numerator and the denominator equal 18, making this step straightforward. When a fraction's numerator and denominator are equal, the fraction simplifies to 1.
This means the previous fraction \( \frac{18}{18} \) simplifies to 1. "1" is the simplest form since it represents an identity element in multiplication, meaning that multiplying anything by 1 does not change its identity.

Distributive Property

The distributive property is a helpful algebraic tool which allows you to multiply a single term by terms contained in parentheses. In our example, the term outside the parentheses \( \frac{9}{2} \) is distributed, or multiplied with, each term inside \( \frac{2}{9} x \).

• First, multiply \( \frac{9}{2} \) with \( \frac{2}{9} x \), following the fraction multiplication rules.
• After multiplying, the resulting expression is \( 1 \times x = x \).

The process of distributing and then simplifying leads to a cleaner, more manageable expression.
It's an essential operation that not only simplifies problems but also sets a strong foundation for more advanced algebraic operations.