Question

Simplify each expression. Assume that all variables are positive when they appear. $$(\sqrt{3}+3)(\sqrt{3}-1)$$

Short answer

The simplified expression is \(2\sqrt{3}\).

Step-by-step solution

Identify the expressions in the product

The given expression is \((\sqrt{3} + 3)(\sqrt{3} - 1)\). Identify the two binomials that need to be multiplied together.

Apply the distributive property

Use the distributive property (also known as FOIL method) to multiply each term in the first binomial by each term in the second binomial. This results in \((\sqrt{3})(\sqrt{3}) + (\sqrt{3})(-1) + (3)(\sqrt{3}) + (3)(-1)\).

Simplify the individual products

Simplify each term:\[(\sqrt{3})(\sqrt{3}) = 3\]\[(\sqrt{3})(-1) = -\sqrt{3}\]\[(3)(\sqrt{3}) = 3\sqrt{3}\]\[(3)(-1) = -3\]

Combine like terms

Combine the simplified terms: \(3 - \sqrt{3} + 3\sqrt{3} - 3\).

Final simplification

Combine the like terms to simplify the expression: \(3 - 3 - \sqrt{3} + 3\sqrt{3} = 0 + 2\sqrt{3} = 2\sqrt{3}\). The final simplified expression is \(2\sqrt{3}\).

Key concepts

Understanding the Distributive Property

The distributive property is a crucial algebraic rule. It allows you to multiply a single term by each term inside a set of parentheses. In general, it states that \textbf{a(b + c) = ab + ac}. Imagine distributing the multiplication over each term within the parentheses.
In the given problem \((\text{√3 } + 3)(\text{√3 } - 1)\), we distribute \(\text{√3}\text{ and } 3\) to both \( \text{√3 }\text{ and } -1 \). Doing this step gives us individual terms:

• \text{√3 } \text{× √3}
• \text{√3 } \text{× (-1)}
• 3 \text{× √3}
• 3 \text{× (-1)}

Breaking them apart helps in simplifying each term before moving forward.

Using the FOIL Method

FOIL stands for First, Outside, Inside, Last. This method is especially handy when multiplying binomials. It makes sure you don't miss any term during multiplication.
Applying FOIL to \( (\text{√3 } + 3)(\text{√3 } - 1) \):

• \textbf{First: }Multiply the first terms of each binomial: \( \text{√3 }\text{×√3} = 3 \)
• \textbf{Outside: }Multiply the outer terms: \( \text{√3}\text{× (-1)} = -\text{√3} \)
• \textbf{Inside: }Multiply the inner terms: \( 3 \text{× } \text{√3} = 3\text{√3} \)
• \textbf{Last: }Multiply the last terms: \( 3 \text{× (-1)} = -3 \)

By following FOIL, you ensure all terms from the product are accounted for.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. Like terms have identical variable parts raised to the same power.
In the expression \ (3 - \text{√3 } + 3\text{√3 } - 3) \, group similar terms to simplify:

• Combine constants: \(3 - 3 = 0\)
• Combine radical terms: \(-\text{√3 } + 3\text{√3 } = 2\text{√3}\)

The final simplified term is \ (0 + 2\text{√3 } = 2\text{√3}) \. This process makes expressions simpler and more manageable.