Problem 57 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ 6(\sqrt{2}-2) $$

Step-by-Step Solution

Answer

The expression simplifies to \(6\sqrt{2} - 12\).

Step 1: Distribute the Multiplier

To solve the expression \(6(\sqrt{2} - 2)\), begin by distributing the 6 to both terms inside the parentheses. This means you multiply 6 by \(\sqrt{2}\) and 6 by -2.

Step 2: Multiply Terms Individually

First, multiply 6 by \(\sqrt{2}\) to get \(6\sqrt{2}\). Then, multiply 6 by -2 to get -12. This results in the expression \(6\sqrt{2} - 12\).

Step 3: Simplify the Expression

Check if the expression \(6\sqrt{2} - 12\) can be simplified further. Since \(6\sqrt{2}\) and -12 are not like terms, this is the simplest form of the expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is an important part of algebra, as it makes equations easier to understand and solve. To simplify an expression, you combine like terms and apply mathematical operations to simplify the overall equation. In the expression \(6(\sqrt{2} - 2)\), simplifying involves using the distributive property to remove the parentheses.The distributive property states that you multiply the number outside the parentheses by each term within the parentheses. For this expression, you distribute the number 6. Each term is dealt with separately, and in this case, results in \(6\sqrt{2} - 12\).

• Distributing ensures you fully multiply each term inside the parentheses by the external factor.
• Once distributed, you check if further simplification can occur. This might include combining like terms or reducing fractions where possible.

This expression has no further like terms to combine, so you reach the simplest form right after distribution.

Multiplying Radicals

Understanding radicals, or root expressions, is crucial when simplifying expressions that involve them. When multiplying radicals, you can often simplify by multiplying the radicands—the numbers inside the radical symbol.In the expression \(6(\sqrt{2} - 2)\), 6 is not a radical, but you apply multiplication to a radical term: \(6\sqrt{2}\). Here’s how you handle it:

• The 6 is multiplied directly to the radical, resulting in \(6\sqrt{2}\). You essentially treat the entire radical as a single unit rather than altering the number under the square root.

Therefore, \(6\sqrt{2}\) stays as it is because \(6\) is an integer and \(\sqrt{2}\) is already in simplified form. Always verify whether a radical can be simplified, but in this instance, \(\sqrt{2}\) is already simplified since it cannot be broken down further without becoming irrational.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra, consisting of numbers, variables, and arithmetic operations. The example \(6(\sqrt{2} - 2)\) is an algebraic expression combining constants and radicals.

• Constants, like -2 and 6, are numbers without variables.
• Radicals, like \(\sqrt{2}\), involve roots. Combined with numbers, they form non-linear expressions.

In manipulating algebraic expressions, you often rely on properties like distribution to rewrite or simplify them. The distribution allowed us to express \(6(\sqrt{2} - 2)\) as \(6\sqrt{2} - 12\). Understanding each component helps break down complex expressions into manageable parts, making them easier to solve or simplify further.By grasping how to manipulate algebraic expressions, you'll be equipped to handle more complex algebraic operations that build on these foundational skills.