Problem 57 - Algebra A Combined Function

Question

Multiply. Then simplify if possible. Assume that all variables represent positive real numbers. $$ 6(\sqrt{2}-2) $$

Step-by-Step Solution

Answer

The simplified expression is \( 6\sqrt{2} - 12 \).

Step 1: Understand the Expression

The expression given is \( 6(\sqrt{2} - 2) \). It indicates that we need to distribute the 6 across each term inside the parentheses.

Step 2: Distribute the 6

We multiply the 6 by each term inside the parentheses: \( 6 \times \sqrt{2} - 6 \times 2 \). This results in \( 6\sqrt{2} - 12 \).

Step 3: Simplify (if possible)

Check if there are any common factors that can simplify the expression further. In this case, \( 6\sqrt{2} - 12 \) does not have any common factors, so it is already in its simplest form.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, which allows us to simplify expressions by distributing a multiplier across terms inside parentheses. In the expression \( 6(\sqrt{2} - 2) \), we apply the distributive property by multiplying 6 with each term in the bracket, separately.

This operation can be broken down as follows:

• First, multiply 6 by \( \sqrt{2} \), which results in \( 6\sqrt{2} \).
• Next, multiply 6 by -2, giving us -12.

By using the distributive property, we transform the expression into \( 6\sqrt{2} - 12 \), which is easier to work with for further calculations. Remember, the key is to "distribute" the number outside the parentheses to each number inside it, and then perform the standard operations.

Simplification

Simplification is the process of rewriting an expression in its most reduced form, making it easier to understand or further manipulate mathematically. After applying the distributive property, we obtained the expression \( 6\sqrt{2} - 12 \).

This expression is already simplified because there are no like terms or common factors that can further reduce it. In algebra, like terms are those that have the same variables raised to the same power, but in this case, neither \( 6\sqrt{2} \) nor \(-12\) share any commonalities beyond multiplication. Therefore, no further simplification is possible beyond recognizing this expression as its simplest form.

Always verify if further simplification is possible by checking for common factors or like terms. Simplifying makes mathematical expressions easier to handle in future algebraic operations.

Square Roots

The square root is a mathematical operation that finds a number, which when multiplied by itself, yields the original number under the root symbol. In our expression, we see \( \sqrt{2} \), which represents a number that, when squared, equals 2.

Square roots often appear in algebra, and they're crucial for solving equations that involve quadratic relationships. When performing algebraic operations involving square roots:

• Treat the square root as you would any other variable or constant during multiplication and addition.
• Be aware that the product of two square roots can be simplified into a single square root, e.g., \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \).

In our case, during simplification, note that while \( \sqrt{2} \) is a constant irrational number, it behaves like a variable in this context and swiftly integrates into the multiplication using the distributive property.

This unique attribute makes understanding square roots an essential part of mastering algebra and other related fields.