Problem 29 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ \sqrt{24}-2 \sqrt{54}-4 \sqrt{12} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: $$\sqrt{24} - 2\sqrt{54} - 4\sqrt{12}$$ Answer: $$-4\sqrt{6} - 8\sqrt{3}$$

Step 1: Break down the square roots into their simplest radical form

To break down the square roots into their simplest radical form, find the factors of the numbers inside the square roots. For each number, try to find the largest perfect square that is a factor of the number. $$ \sqrt{24} = \sqrt{4 \cdot 6} = 2\sqrt{6} $$ $$ \sqrt{54} = \sqrt{9 \cdot 6} = 3\sqrt{6} $$ $$ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} $$ Now, rewrite the given expression with the simplified radicals: $$ 2\sqrt{6} - 2 \cdot 3\sqrt{6} - 4 \cdot 2\sqrt{3} $$

Step 2: Combine like terms

Now that we have broken down the square roots, look for any like terms that can be combined. In this expression, we can combine the terms with square roots of 6: $$ 2\sqrt{6} - 6\sqrt{6} = -4\sqrt{6} $$ And the term with the square root of 3 will remain unchanged: $$ -4 \cdot 2\sqrt{3} = -8\sqrt{3} $$ Now, put the expression back together: $$ -4\sqrt{6} - 8\sqrt{3} $$

Step 3: Final Answer

The simplified expression is: $$ -4\sqrt{6} - 8\sqrt{3} $$

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, or other roots. Here, we focus on square roots, identified with the symbol \( \sqrt{} \). A square root of a number \( x \) is a number \( y \) such that \( y^2 = x \). For example, \( \sqrt{16} = 4 \) because \( 4^2 = 16 \).

To simplify radical expressions, it’s important to look for perfect square factors. A **perfect square** is a number whose square root is an integer like 1, 4, 9, 16, and so on. For instance, in the expression \( \sqrt{24} \), we can break it down using \( 4 \), a perfect square, to obtain \( 2\sqrt{6} \). This process involves:

• Identifying the largest perfect square factor of the radicand (the number inside the square root).
• Rewriting the square root using this perfect square.
• Simplifying to separate the perfect square’s square root from the rest (e.g., \( \sqrt{4} = 2 \)).

This process helps in simplifying the expression, making it easier to work with.

Combining Like Terms

Combining like terms is a fundamental algebraic process. It involves adding or subtracting terms with the same variables raised to the same power. This can be especially useful when simplifying expressions.

In our example, the goal was to combine terms with the same radical part. We first simplified terms to **like terms** such as \( 2\sqrt{6} \), \( 6\sqrt{6} \), and \( 8\sqrt{3} \). Then, we combined the terms with \( \sqrt{6} \):

• Simplify coefficients separately (e.g., \( 2\sqrt{6} - 6\sqrt{6} = -4\sqrt{6} \)).
• Non-similar terms remain unchanged and are simply carried along (e.g., \( -8\sqrt{3} \) stays the same).

This step ensures the expression is in its simplest form, making further operations more straightforward.

Perfect Squares

Perfect squares play a vital role when dealing with radical expressions. They are numbers like 1, 4, 9, 16, 25, which can be expressed as the square of an integer (\( n^2 \)).

When simplifying, **finding perfect square factors** is crucial. For instance, in \( \sqrt{54} \):

• Recognize \( 9 \) as a perfect square factor since \( 9 = 3^2 \).
• Rewrite \( \sqrt{54} = \sqrt{9 \cdot 6} \) to simplify as \( 3\sqrt{6} \).

This method not only simplifies the expression but also lays the groundwork for combining like terms. Recognizing perfect squares allows for break down into simpler, manageable pieces, vital for most algebraic simplifications.