Problem 322 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{3+\sqrt{27}}{9} $$

Step-by-Step Solution

Answer

\( \frac{1+\text{\root{(3)}}}{3} \)

Step 1: Identify Terms

The expression to simplify is \( \frac{3+\frac{\text{\root{(27)}}}}{9} \text{.} \). The numerator is \( 3+\text{\root{(27)}} \).

Step 2: Simplify the Radical

Simplify \( \text{\root{(27)}} \). Since \( 27 = 9 \times 3 \), we have \( \text{\root{(27)}} = \text{\root{(9 \times 3)}} = \text{\root{(9) \times \text\root{(3)}}} = 3 \times \text{\root{(3)}} = 3 \text{\root{(3)}} \). Thus, the expression becomes \( \frac{3 + 3\text{\root{(3)}}}{9} \).

Step 3: Factor Out the Common Term

Factor out the common term in the numerator, which is 3: \( \frac{3(1 + \text{\root{(3)}})}{9} \).

Step 4: Simplify the Fraction

Divide the numerator by the denominator: \( \frac{3(1 + \text{\root{(3)}})}{9} = \frac{1+\text{\root{(3)}}}{3} \).

Key Concepts

simplifying radicals

When simplifying radicals, it's important to identify the square roots within the expression. For instance, with \(\text{\root{(27)}}\), we can break it down into its prime factors: \(27 = 9 \times 3\). We know that \(9\) is a perfect square, so we can simplify: \[ \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3 \sqrt{3} \]. Always check if the radicand (the number under the root) can be decomposed into smaller factors, especially perfect squares, to simplify your work.

factoring out common terms

Factoring out common terms involves identifying and extracting common elements from within an expression. In our example, the numerator \(3 + 3\sqrt{3}\) has a common factor of 3: \[ 3 + 3\sqrt{3} = 3(1 + \sqrt{3}) \]. By doing this, we simplify the expression, making it easier to work with. Remember, the goal is to make both the numerator and denominator as simple as possible for any further operations.

division in fractions

Understanding division in fractions is essential for simplifying expressions. Once you've factored out the common term in the numerator, as in our example \( \frac{3(1+\sqrt{3})}{9}\), you can then divide both the numerator and the denominator by the greatest common divisor. Here, since both 3 and 9 have the greatest common divisor of 3, it becomes: \[ \frac{3(1+\sqrt{3})}{9} = \frac{1+\sqrt{3}}{3} \]. This results in a simplified expression that's easier to understand and work with.

step-by-step problem solving

Solving algebraic problems step by step ensures clarity and accuracy. Here's a detailed rundown of our process:
• Identify all terms within the expression.
• Simplify any radicals present.
• Factor out common terms to simplify the numerator or denominator.
• Divide the numerator by the denominator by using their greatest common divisor.
Applying this method consistently enhances your problem-solving skills, making complex expressions manageable and ensuring no steps are skipped.