Problem 73 - Elementary Algebra

Question

\(6\left(\frac{2}{3} x+\frac{4}{9}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(4x + \frac{8}{3}\).

Step 1: Distribute 6 to Each Term Inside the Parentheses

Distribute the 6 to both \(\frac{2}{3} x\) and \(\frac{4}{9}\). Use the distributive property: \(a(b + c) = ab + ac\). Thus, \(\begin{align*} 6\bigg(\frac{2}{3} x\bigg) + 6\bigg(\frac{4}{9}\bigg) \).

Step 2: Simplify Each Term

Simplify both terms by multiplying: \(\begin{align*} 6 \times \frac{2}{3}x &= 4x \ 6 \times \frac{4}{9} &= \frac{24}{9} = \frac{8}{3} \).

Step 3: Write the Simplified Expression

Combine both simplified terms: \(4x + \frac{8}{3}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra. They are combinations of variables, numbers, and operators (like +, -, *, /). Expressions can be very simple, like 'x + 2', or more complex, like the one in our example: \( 6\left(\frac{2}{3} x+\frac{4}{9}\right) \). When working with algebraic expressions, we often need to simplify them to make them easier to understand or solve.

Understanding and practicing how to manipulate algebraic expressions is crucial in algebra. Using the distributive property, we can break down and simplify these expressions more effectively. Remember the basic structure of algebraic expressions and how each component interacts with the others.

Simplification

Simplification is all about making expressions easier to work with. It means reducing an expression to its simplest form. Let's look at our example: \( 6\left(\frac{2}{3} x+\frac{4}{9}\right) \).

We first used the distributive property in this problem: \(a(b + c) = ab + ac\). By applying this property, we distributed 6 to both terms inside the parentheses, leading to:
\( 6\left(\frac{2}{3}x\right) + 6\left(\frac{4}{9}\right) \)

Next, we performed the multiplication to simplify each term. For the term \(6\left(\frac{2}{3}x\right)\), the result was \(4x\). And for \(6\left(\frac{4}{9}\right)\), it simplified to \( \frac{8}{3} \). Finally, combining these results, we get the simplified expression: \( 4x + \frac{8}{3} \).

Following these steps ensures a clear and accurate simplification process, making it easier to work with the expression later on.

Multiplication of Fractions

Handling the multiplication of fractions accurately is essential in algebra. Here’s a detailed look at how to do it using our example.

When multiplying fractions, multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For instance, in \(6\left(\frac{2}{3} x\right)\), treat 6 like a fraction: \(\frac{6}{1}\). Then multiply: \( \frac{6}{1} \times \frac{2}{3} = \frac{6 \times 2}{1 \times 3} = \frac{12}{3} = 4 \). The same method applies to our second term: \( \frac{6}{1} \times \frac{4}{9} = \frac{6 \times 4}{1 \times 9} = \frac{24}{9} = \frac{8}{3} \).

Always remember to simplify fractions if possible. Simplification makes your work cleaner and often easier to understand.
By following these steps, you can handle any fraction multiplication problem you encounter in algebra.