Question

Use the Distributive Property to remove the parentheses. $$ 3\left(\frac{2}{3} x+\frac{1}{6}\right) $$

Short answer

The expression simplifies to \2x + \frac{1}{2}\.

Step-by-step solution

Identify the expression inside the parentheses

The expression inside the parentheses is \[\frac{2}{3} x + \frac{1}{6}\].

Multiply each term inside the parentheses by the factor outside

Use the distributive property to multiply 3 with each term inside the parentheses. This gives: \[\begin{array}{l} 3 \times \frac{2}{3} x + 3 \times \frac{1}{6} \end{array}\]

Simplify each term

Simplify the expression term by term: \[\begin{array}{l} 3 \times \frac{2}{3} x = 2x \end{array} \] and \[\begin{array}{l} 3 \times \frac{1}{6} = \frac{1}{2} \end{array} \]

Combine the simplified terms

Put the simplified terms together to get: \[\begin{array}{l} 2x + \frac{1}{2} \end{array}\]

Key concepts

Simplifying Expressions

Simplifying expressions is a core skill in algebra that involves rewriting an expression in a more compact or understandable form. Simplification can involve various steps:

• Removing parentheses
• Combining like terms
• Multiplying out factors

In the given exercise, we start with the expression \[ 3\bigg(\frac{2}{3} x + \frac{1}{6}\bigg) \]. By employing the Distributive Property, we simplify this into: \[ 2x + \frac{1}{2}. \]
This step-by-step breakdown not only makes the expression easier to understand but also prepares it for further algebraic manipulations.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions to make them easier to work with. This includes adding, subtracting, multiplying, and dividing terms. In our exercise:

• We identified the expression inside the parentheses: \[\frac{2}{3} x + \frac{1}{6}\]
• We used the distributive property to multiply 3 with each term inside the parentheses.

By doing so, we transformed \[ 3 \bigg(\frac{2}{3} x + \frac{1}{6} \bigg) \] into \[ 3 \times \frac{2}{3} x + 3 \times \frac{1}{6} \], which simplifies to \[ 2x + \frac{1}{2} \].
These steps are crucial for making difficult problems more manageable and for ensuring that the final expression is as simple as possible.

Fraction Operations

Understanding fraction operations is essential when dealing with algebraic expressions that include fractions. Here are the key points covered in our exercise:

• Multiplying fractions: We multiplied \[ 3 \times \frac{2}{3} \] and \[ 3 \times \frac{1}{6} \].
• Simplifying fractions: \[ 3 \times \frac{2}{3} = 2 \] and \[ 3 \times \frac{1}{6} = \frac{1}{2} \].

By breaking down these operations, we can simplify the terms independently. When multiplying fractions, you multiply the numerators (top numbers) and the denominators (bottom numbers) separately. Then, reduce the resulting fractions to their simplest form.

Combining Like Terms

Combining like terms is another fundamental skill in algebra that aids in simplifying expressions. Like terms are terms that contain the same variables raised to the same power.

• In the expression \( 2x + \frac{1}{2} \), 2x and \(\frac{1}{2}\) are not like terms because they do not share the same variable part. Therefore, they cannot be combined into a single term but are placed together in the final expression.

Combining like terms helps to reduce the number of terms in an expression, making it simpler and easier to work with. In our case, we arrived at the simplest form of the expression: \[ 2x + \frac{1}{2} \]