Problem 57 - Intermediate Algebra

Question

Simplify by combining like radicals. $$ 4+\sqrt{8}+\sqrt{2}+8 $$

Step-by-Step Solution

Answer

The simplified expression is \( 12 + 3\sqrt{2} \).

Step 1: Identify Like Radicals

First, we observe the expression: \[ 4 + \sqrt{8} + \sqrt{2} + 8 \] Identify any like radicals in the expression. Like radicals have the same radicand. Here, \( \sqrt{8} \) and \( \sqrt{2} \) are the radical terms.

Step 2: Simplify Radicals

Simplify the radical expressions where possible. The term \( \sqrt{8} \) can be simplified: \[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \] Thus, the expression becomes: \[ 4 + 2\sqrt{2} + \sqrt{2} + 8 \]

Step 3: Combine Like Radical Terms

Combine the like radical terms \( 2\sqrt{2} \) and \( \sqrt{2} \): \[ 2\sqrt{2} + \sqrt{2} = 3\sqrt{2} \] Substitute back into the expression, which becomes: \[ 4 + 3\sqrt{2} + 8 \]

Step 4: Combine Non-Radical Terms

Add the non-radical terms (constants) together: \[ 4 + 8 = 12 \] Now, substitute this result back into the expression: \[ 12 + 3\sqrt{2} \] This is the simplified expression.

Key Concepts

Radical Expressions

Radical expressions are expressions that contain a square root, cube root, or higher-order roots. These expressions often pose a challenge when solving algebraic equations. The symbol \( \sqrt{} \) represents the square root, while an expression like \( \sqrt[3]{} \) signifies the cube root. Understanding the properties of these roots is key to working efficiently with radicals.

In the expression \( \sqrt{8} + \sqrt{2} \), each component is a radical expression. The number inside the radical sign is called the radicand. For like radicals, the radicands must be identical for them to be combined directly. This concept closely resembles combining like terms in polynomial expressions, which helps simplify calculations and work towards solving the equation.

Simplifying Radicals

Simplifying radicals involves breaking down a radical expression into its simplest form. This often means expressing it in terms of its prime factors, and then extracting any perfect squares from under the radical sign. For example, consider simplifying \( \sqrt{8} \).

• First, factor the number under the square root: \( 8 = 4 \times 2 \).
• Recognize that \( 4 \) is a perfect square: \( \sqrt{4} = 2 \).
• Express \( \sqrt{8} \) as \( \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \).

This simplification process makes it easier to identify and combine like radicals effectively. Always simplify each radical term individually before combining to maintain accuracy in the solution.

Algebraic Simplification

Algebraic simplification involves reducing an expression to its simplest or most efficient form. This process includes combining like terms, factoring, and reducing fractions. Simplification makes it easier to interpret and solve algebraic equations.

In the given exercise, the expression \( 4 + \sqrt{8} + \sqrt{2} + 8 \) becomes \( 12 + 3\sqrt{2} \) through a series of simplification steps:

• Simplify radicals individually, as seen with \( \sqrt{8} = 2\sqrt{2} \).
• Combine like radicals: \( 2\sqrt{2} + \sqrt{2} = 3\sqrt{2} \).
• Combine constant terms: \( 4 + 8 = 12 \).

The expression is now more straightforward and quite manageable, showcasing the power of algebraic simplification in solving complex problems.