Problem 78 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{2 a+1}+\sqrt{8 a+4} $$

Step-by-Step Solution

Answer

The simplified expression is \(3\sqrt{2a+1}\).

Step 1: Factorize the Expression Inside the Square Root

Start by factorizing the expressions inside the square roots where you can. For \( \sqrt{8a + 4} \), notice that you can factor out a 4.\[ 8a + 4 = 4(2a + 1) \] So now we have the expression: \[ \sqrt{2a + 1} + \sqrt{4(2a + 1)} \]

Step 2: Simplify the Second Square Root

The next step is to simplify \( \sqrt{4(2a + 1)} \). Use the property of square roots that \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \).Applying this, we get:\[ \sqrt{4(2a + 1)} = \sqrt{4} \cdot \sqrt{2a + 1} \]Since \( \sqrt{4} = 2 \), this becomes:\[ 2 \cdot \sqrt{2a + 1} \]

Step 3: Combine Like Terms

Now rewrite the original expression using the simplified form:\[ \sqrt{2a + 1} + 2\sqrt{2a + 1} \]Notice both terms have the same square root, allowing them to be combined as like terms:\[ (1 + 2)\sqrt{2a + 1} = 3\sqrt{2a + 1} \]

Step 4: Write the Final Simplified Expression

After combining the terms, the expression simplifies to:\[ 3\sqrt{2a + 1} \] This is the simplest form of the original expression.

Key Concepts

Simplifying Expressions

Simplifying an algebraic expression often involves a few key steps to transform it into its simplest form. The goal is to rewrite the expression in a more efficient way that is easier to work with. Here are some steps you might take:

• Look for common factors that can be factored out. This can make subsequent steps easier.
• Simplify any fractions by finding the greatest common divisor for the numerator and denominator.
• Combine like terms, such as terms with the same variable part, to minimize redundancy.
• Use properties of operations, like distributive or associative properties, to rearrange and simplify expressions.

For example, in the original exercise, the expression involves square roots and requires different strategies like factoring and combining like terms to reach the simplest form.

Square Roots

Square roots are mathematical operations that find the original number that was squared to get a given number. The symbol \( \sqrt{} \) represents the square root. When working with square roots in algebra, you often need to simplify them:

• Factor the number inside the square root into a product of perfect squares and other factors.
• Apply the property \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \), to split the square root into manageable parts.

For instance, in the exercise, the expression \( \sqrt{8a + 4} \) is simplified by factoring it to \( 4(2a + 1) \). The square root can then be split using its properties, simplifying the expression to \( 2 \cdot \sqrt{2a + 1} \). This helps in making the expression more straightforward.

Factoring

Factoring is an essential process in algebra where you express a mathematical expression as a product of its factors. It's like reverse multiplying. Here's how you can factor expressions:

• Identify common factors in the terms. If multiple terms share a common factor, you can factor it out.
• Look for patterns, like a difference of squares or trinomials, which can be factored using special formulas.

In our exercise, we factored the expression inside the square root \( 8a + 4 \) by pulling out the common factor 4, resulting in \( 4(2a + 1) \). This makes it easier to simplify the square root that follows, demonstrating the usefulness of factoring in simplifying complex expressions.

Like Terms

Like terms are terms within an expression that have identical variable parts raised to the same power. Simplifying an expression includes combining these like terms to minimize complexity. When you identify like terms:

• Check that the variable portion is exactly the same in each term.
• Sum or subtract the coefficients of these terms.

In the final steps of the exercise, once the square roots were simplified, the terms \( \sqrt{2a + 1} \) and \( 2\sqrt{2a + 1} \) were combined because they share the same square root. This results in \( (1 + 2)\sqrt{2a + 1} = 3\sqrt{2a + 1} \). This simplification process is crucial to solving algebraic expressions efficiently.