Problem 102 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 27\left(\frac{2}{3} x\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(18x\).

Step 1: Distribute the Number Outside the Parentheses

First, observe the expression: \(27\left(\frac{2}{3} x\right)\). This means we need to multiply 27 by each term within the parentheses. Since there is only one term, \(\frac{2}{3} x\), we'll multiply 27 by that.

Step 2: Multiply the Coefficients

Calculate the multiplication of the coefficients, which are 27 and \(\frac{2}{3}\). To do this, multiply 27 by 2 and then divide the result by 3.\[ 27 \times \frac{2}{3} = \frac{27 \times 2}{3} = \frac{54}{3} \]

Step 3: Simplify the Resulting Fraction

Now simplify the fraction \(\frac{54}{3}\). Divide 54 by 3:\[ \frac{54}{3} = 18 \]

Step 4: Final Expression

Combine the simplified coefficient with the variable \(x\):\[ 18x \]

Key Concepts

Expression Simplification

When we talk about expression simplification, we're referring to the process of making a mathematical expression easier to work with. This involves reducing it to its simplest form while keeping the same value. Let's break it down using the example provided.

• The given expression is \(27\left(\frac{2}{3} x\right)\). Our goal is to simplify it.
• The first step is to focus on the part inside the parentheses. In this case, it's \(\frac{2}{3} x\).
• By simplifying, we aim to remove any unnecessary parts and make calculations easier.

In terms of arithmetic operations, simplification often involves reducing fractions, like converting \(\frac{54}{3}\) to 18. This step-by-step breakdown helps ensure that you grasp the operations you need.

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a number by a sum or difference inside parentheses. It's expressed as \(a(b + c) = ab + ac\). This property can be very handy when dealing with expressions like ours. Let's see it in action.

• Look at the expression \(27\left(\frac{2}{3} x\right)\). Notice the number 27 outside the parentheses.
• According to the distributive property, we need to multiply 27 by every term inside the parentheses. Here, there's only one term: \(\frac{2}{3} x\).
• Effectively, you're distributing the number outside to each term within the parentheses to simplify the expression.

This property is a tool that helps ease the multiplication process, making it simpler to manage larger expressions.

Multiplication of Fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. Here's how it applies to our exercise:

• Our expression requires multiplying 27 by \(\frac{2}{3}\).
• Think of 27 as \(\frac{27}{1}\) to visualize the process more clearly.
• The multiplication becomes \(\frac{27}{1} \times \frac{2}{3}\), which leads to multiplying the numerators \(27 \times 2 = 54\) and the denominators \(1 \times 3 = 3\).
• We end up with \(\frac{54}{3}\).

Simplifying the resultant fraction gives us \(18\). Remember, simplifying fractions often makes them more understandable and easier to work with. The multiplication of fractions is a crucial step in transforming and simplifying expressions.