Problem 8 - Algebra for College Students

Question

Use the distributive property to help simplify each of the following. \(6 \sqrt{12}+\sqrt{3}-2 \sqrt{48}\)

Step-by-Step Solution

Answer

The simplified expression is \(5\sqrt{3}\).

Step 1: Simplify Each Square Root

Begin by simplifying each square root in the expression:\[ 6 \sqrt{12} + \sqrt{3} - 2\sqrt{48} \] First, note that:\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \] Similarly, simplify \( \sqrt{48} \):\[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \] Now rewrite the original expression using these simplifications:

Step 2: Substitute the Simplified Roots

Substitute the simplified square roots into the expression:\[ 6(2\sqrt{3}) + \sqrt{3} - 2(4\sqrt{3}) \] This becomes:\[ 12\sqrt{3} + \sqrt{3} - 8\sqrt{3} \]

Step 3: Combine Like Terms

Identify and combine the like terms. All terms include \( \sqrt{3} \), so add and subtract the coefficients: \[ 12\sqrt{3} + \sqrt{3} = 13\sqrt{3} \] \[ 13\sqrt{3} - 8\sqrt{3} = 5\sqrt{3} \]

Step 4: Write the Final Simplified Expression

The expression simplifies to:\[ 5\sqrt{3} \] This is the final simplified form of the original expression.

Key Concepts

Simplifying Square Roots

When dealing with square roots in algebraic expressions, simplifying them can make the problem easier to tackle. A square root is considered simplified when there are no perfect square factors left under the root. For example, \( \sqrt{12} \) can be broken down by identifying that 12 is a product of 4 and 3. Since 4 is a perfect square, you can write \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} \). Knowing that \( \sqrt{4} = 2 \), the expression further simplifies to \( 2\sqrt{3} \). Similarly, \( \sqrt{48} \) can be broken down into \( \sqrt{16 \times 3} \), simplifying to \( 4\sqrt{3} \) because \( \sqrt{16} = 4 \). This technique reduces complicated square roots into simpler, more manageable forms.

Combining Like Terms

After simplification, the next step in solving algebraic expressions is to combine like terms, which essentially means simplifying the expression by collecting similar elements. Like terms are terms that have the same variable raised to the same power. In this case, we worked with several terms all having \( \sqrt{3} \) as a factor. Consider the expression:

• \( 12\sqrt{3} \)
• \( +\sqrt{3} \)
• \( -8\sqrt{3} \)

You can combine these by simply adding and subtracting their coefficients:

• \( 12\sqrt{3} + \sqrt{3} = 13\sqrt{3} \)
• \( 13\sqrt{3} - 8\sqrt{3} = 5\sqrt{3} \)

By efficiently combining the terms, we arrive at a much simpler and cleaner expression: \( 5\sqrt{3} \).

Algebraic Expressions

Algebraic expressions like \( 6 \sqrt{12} + \sqrt{3} - 2 \sqrt{48} \) are made up of variables, numbers, and operational symbols. They are simplified using a combination of mathematical rules including the distributive property, simplifying square roots, and combining like terms.

An important concept is to first simplify the components, such as square roots, which makes combining terms more straightforward. These expressions are a fundamental part of algebra and help represent real-world problems in a mathematical format.

Learning how to manipulate and simplify algebraic expressions is crucial because they form the basis for solving equations and understanding functions. By breaking down these expressions into simpler parts, you make the problem-solving process clearer and more efficient.