Problem 77 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{4 x+8}+\sqrt{x+2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(3\sqrt{x+2}\).

Step 1: Identify Common Factors

Observe the terms inside the square root: \(\sqrt{4x + 8}\) and \(\sqrt{x + 2}\). Notice that 8 can be expressed as \(4 \times 2\), indicating that \(4x + 8\) has a common factor of 4.

Step 2: Factor the Expression

Write the expression inside the first square root in its factored form: \(\sqrt{4(x + 2)}\). Now the expression is \(\sqrt{4(x+2)} + \sqrt{x+2}\).

Step 3: Simplify Using Properties of Square Roots

Use the property that \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\). Therefore, \(\sqrt{4(x+2)} = \sqrt{4} \cdot \sqrt{x+2} = 2\sqrt{x+2}\).

Step 4: Combine Like Terms

Replace \(\sqrt{4(x+2)}\) with \(2\sqrt{x+2}\) in the expression: \(2\sqrt{x+2} + \sqrt{x+2}\). Since both terms are like terms, combine them: \(2\sqrt{x+2} + 1\sqrt{x+2} = 3\sqrt{x+2}\).

Step 5: Write the Simplified Expression

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

Key Concepts

Square Roots

Square roots are a way to determine a number that, when multiplied by itself, results in the original number. For instance, the square root of 9 is 3 because 3 times 3 equals 9.
When dealing with square roots in algebraic expressions, you might encounter square roots of variables or terms.
For example, in the expression \( \sqrt{4x + 8} \), you're taking the square root of the whole expression inside the radical sign.

• Square roots simplify expressions by factoring numbers within the root into smaller parts.
• Understanding square roots helps simplify algebraic expressions involving radical terms.

Factoring

Factoring is breaking down a complex expression into simpler components, known as factors, which, when multiplied together, result in the original expression. This is crucial when simplifying expressions under a square root.
When you examine \( 4x + 8 \), observe that both terms have a common factor: 4. So, you can rewrite the expression as \( 4(x + 2) \). This makes it simpler to work with, especially under a square root.
Breaking expressions down into factors allows you to apply other algebraic rules effectively.

• Factoring simplifies expressions inside square roots.
• Helps in reducing complex expressions to easier forms to manipulate.

Like Terms

Like terms are terms within an expression that have the same variable(s) raised to the same power, allowing them to be combined or added together.
For example, in the expression \( 2\sqrt{x+2} + \sqrt{x+2} \), both terms involve \( \sqrt{x+2} \).
This makes them like terms, meaning you can sum them up because they share the same variable expression within the root. In this case, \( 2\sqrt{x+2} + 1\sqrt{x+2} = 3\sqrt{x+2} \).

• Like terms facilitate the simplification of expressions by allowing terms to be added directly.
• Recognizing like terms is key in combining and reducing expressions.

Properties of Square Roots

The properties of square roots are rules that help in manipulating and simplifying expressions involving square roots. One of the most handy properties is when \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \), which allows you to separate a square root into its factors.
In the equation \( \sqrt{4(x+2)} \), this property allows for simplification as \( \sqrt{4} \cdot \sqrt{x+2} \). Because \( \sqrt{4} = 2 \), this expression becomes \( 2\sqrt{x+2} \).
These properties are essential tools that enable you to simplify complex algebraic expressions efficiently.

• Properties allow for separation and simplification of terms under a radical.
• Understanding these properties aids in effective problem-solving in algebra.