Problem 31 - Elementary and Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ -2 \sqrt{5 x}(4 \sqrt{2 x}-3 \sqrt{3}) $$

Step-by-Step Solution

Answer

The simplified form is \(-8x \sqrt{10} - 6 \sqrt{15x}\).

Step 1: Apply the Distributive Property

To solve the problem, start by applying the distributive property, which involves multiplying each term within the parenthesis by \(-2 \sqrt{5x}\). Set up the expression as follows:\[-2 \sqrt{5x} \times 4 \sqrt{2x} - 2 \sqrt{5x} \times 3 \sqrt{3}\]

Step 2: Simplify the First Term

Now, simplify each part of the expression separately. Begin with the first term:\[-2 \times 4 \times \sqrt{5x} \times \sqrt{2x} = -8 \times \sqrt{(5x)(2x)}\]Combine the terms inside the square root:\[-8 \times \sqrt{10x^2} = -8 \times x \times \sqrt{10}\]This results in:\[-8x \sqrt{10}\]

Step 3: Simplify the Second Term

Now, simplify the second term similarly:\[-2 \times 3 \times \sqrt{5x} \times \sqrt{3} = -6 \times \sqrt{15x}\]Keep the expression as:\[-6 \sqrt{15x}\]

Step 4: Combine Simplified Terms

Combine the results from Step 2 and Step 3 to form the simplified expression:\[-8x \sqrt{10} - 6 \sqrt{15x}\]

Key Concepts

Distributive Property

When we talk about the Distributive Property in algebra, we're essentially discussing a way to manage multiplication over an addition or subtraction operation inside parentheses. It's like unpacking a box where you multiply the term outside the parentheses by each term inside individually.
Imagine you have an expression like \(a(b + c)\) . The Distributive Property tells us that you can multiply \(a\) by both \(b\) and \(c\) separately, giving us \(ab + ac\).
So, in the original exercise, the distributive step looks like this:

• The term \(-2 \sqrt{5x}\) outside the parenthesis is multiplied by each term inside the parentheses, \(4 \sqrt{2x}\) and \(-3 \sqrt{3}\).
• Making it: \(-2 \sqrt{5x} \times 4 \sqrt{2x} - 2 \sqrt{5x} \times 3 \sqrt{3}\).

The trick is to deal with each multiplication one by one, applying the property to simplify your expression and solve it step by step.

Simplification

In algebra, simplification refers to the process of making an expression easier to work with. This involves reducing it to its simplest form. It's all about combining like terms, reducing fractions, and making expressions less cluttered.
From our exercise, after applying the distributive property, we obtain two separate terms which need to be individually simplified:

• The first term \(-2 \times 4 \times \sqrt{5x} \times \sqrt{2x}\) simplifies by first multiplying the constants to get \(-8\), and then combining the terms under the square root to \(\sqrt{10x^2}\). Finally, the square root simplifies to \(x \sqrt{10}\), resulting in \(-8x \sqrt{10}\).
• The second term \(-2 \times 3 \times \sqrt{5x} \times \sqrt{3}\) simplifies to \(-6 \sqrt{15x}\) by combining \(5x\) and \(3\).

In the end, combining these simplified terms gives the final expression: \(-8x \sqrt{10} - 6 \sqrt{15x}\). Simplification helps make the expression clearer and more manageable.

Square Roots

Square roots are a fundamental concept in algebra, representing a value that, when multiplied by itself, gives the original number. The symbol \(\sqrt{}\) denotes a square root.
For instance, \(\sqrt{x} \) means the square root of \(x\).

• In the context of the problem, expressions like \(\sqrt{5x}\) and \(\sqrt{2x}\) are utilized. Multiplying these under the square root involves combining their contents first: \(\sqrt{5x \times 2x} = \sqrt{10x^2}\).
• Notice how \(x^2\) becomes simply \(x\) because \(\sqrt{x^2} = x\).

By understanding and working with square roots, we simplify complex expressions to a more workable form. This is essential in unraveling expressions and is often a crucial step in solving algebraic equations.