Problem 35 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{12 a^{4}} $$

Step-by-Step Solution

Answer

Question: Simplify the radical expression: \(\sqrt{12a^4}\). Answer: The simplified radical expression is \(2a^2\sqrt{3}\).

Step 1: Break down the expression into its prime factors

First, let's break down the number 12 and the variable \(a^4\). The prime factors of 12 are 2, 2, and 3. The variable \(a^4\) can be written as \(a \cdot a \cdot a \cdot a\). So the expression inside the square root becomes: $$ \sqrt{2 \cdot 2 \cdot 3 \cdot a \cdot a \cdot a \cdot a} $$

Step 2: Use the properties of square roots

Recall that for any positive numbers x and y, we have \(\sqrt{xy} = \sqrt{x} \sqrt{y}\). We can now apply this property to the expression inside the square root. We can pair up any factors that are the same and take their square root: $$ \sqrt{2 \cdot2 \cdot 3 \cdot a \cdot a \cdot a \cdot a} = \sqrt{2 \cdot2} \cdot \sqrt{3} \cdot \sqrt{a \cdot a} \cdot \sqrt{a \cdot a} $$

Step 3: Simplify the square roots

Now we can simplify the square roots. Since \(\sqrt{2 \cdot 2} = \sqrt{4} = 2\), the first square root simplifies to 2. For the other square roots, we have \(\sqrt{a \cdot a} = a\). The expression becomes: $$ 2 \cdot \sqrt{3} \cdot a \cdot a $$

Step 4: Rewrite the expression in a simplified form

Finally, we can rewrite the expression to show that we've simplified the radical. The expression becomes: $$ 2a^2\sqrt{3} $$ The simplified radical expression is \(2a^2\sqrt{3}\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number or variable into its prime number components. A prime number is a number greater than 1 that is only divisible by 1 and itself. This process helps simplify complex expressions, especially under a square root sign.

• For the number 12, the prime factorization is 2 × 2 × 3. We list it in terms of its prime numbers to see which parts can form perfect squares.
• Similarly, for a variable like \(a^4\), we express it as \(a \times a \times a \times a \). This identifies perfect squares that simplify inside a radical.

Breaking expressions into prime factors is the first crucial step in simplifying radical expressions. By doing so, we can more easily find components that will yield whole numbers when square rooted.

Properties of Square Roots

Understanding the properties of square roots is essential when simplifying radical expressions. The fundamental properties allow us to manipulate the terms inside the radical effectively. Here are a couple to remember:

• \(\sqrt{xy} = \sqrt{x} \times \sqrt{y}\): This property shows that the square root of a product is the product of the square roots. It is useful when we have prime and other factored terms inside a square root.
• Perfect Squares: If a term like \(x^2\) appears inside a square root, the square root eliminates the square, resulting in \(x\).

Applying these properties helps simplify expressions like \(\sqrt{12a^4}\) into its more digestible parts, especially when dealing with algebraic terms.

Algebraic Simplification

Algebraic simplification involves using properties of numbers and operations to reduce expressions to their simplest form. After prime factorization and applying properties of square roots, simplification resolves the expression fully.
After determining that \(\sqrt{2 \cdot 2}\) results in 2, and \(\sqrt{a \cdot a}\) results in \(a\), we combine these elements.
Using the distributed properties and arithmetic, the terms are reorganized as:

• Combine like terms: \(2 \cdot a \cdot a = 2a^2\).
• Simplify remaining inside square root terms, like \(\sqrt{3}\).

Simplicity and accuracy are the end goals, ensuring that the final result like \(2a^2\sqrt{3}\) is easy to understand.

Radical Expressions

Radical expressions are expressions that contain a square root, cube root, or higher order root. Simplifying these expressions is about reducing their form for clarity and solving processes.
A radical such as \(\sqrt{12a^4}\) contains numbers and variables that can be reduced. It's important to:

• Identify perfect square factors that can be "pulled out," simplifying the expression inside the square root.
• Leave irreducible terms inside the radical to avoid overcomplication.

Approaching radical expressions systematically—by prime factorization, property application, and simplification—leads to a clear, concise mathematical expression in its simplest form.