Problem 3 - Intermediate Algebra

Question

Simplify and reduce each expression. $$ \frac{-6 \pm \sqrt{27}}{3} $$

Step-by-Step Solution

Answer

The simplified form is \(-2 \pm \sqrt{3}\).

Step 1: Simplify the Expression in the Numerator

The expression given is \( \frac{-6 \pm \sqrt{27}}{3} \), which means the numerator is \( -6 \pm \sqrt{27} \). First, simplify \( \sqrt{27} \). Since \( 27 = 9 \times 3 \), \( \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3} \). Hence, the numerator becomes \( -6 \pm 3\sqrt{3} \).

Step 2: Divide Each Term in the Numerator by the Denominator

Now that the numerator is \( -6 \pm 3\sqrt{3} \), we divide each term in the numerator by the denominator, which is 3. This results in: \[ \frac{-6}{3} \pm \frac{3\sqrt{3}}{3} \]. Simplifying each term gives us \( -2 \pm \sqrt{3} \).

Step 3: Combine and Represent the Simplified Expression

The expression simplifies to \( -2 \pm \sqrt{3} \). This means the expression represents two numbers: \(-2 + \sqrt{3}\) and \(-2 - \sqrt{3}\).

Key Concepts

Square Root Simplification

Understanding how to simplify square roots is crucial in algebra. Simplifying \(\sqrt{27}\) begins with breaking down the number 27 into its factors. Specifically, 27 can be expressed as \(9 \times 3\). This step is essential because \(9\) is a perfect square. \(\sqrt{9}\) gives \(3\), so the expression \(\sqrt{27}\) simplifies to \(3\sqrt{3}\). This is a common technique used to simplify square roots:

• Factor the number under the square root into its prime factors or pairs of factors.
• Identify perfect squares among those factors.
• Simplify by taking the square root of the perfect squares.

Finding these simplifications helps make expressions easier to work with in further algebraic operations.

Fraction Reduction

Reducing fractions simplifies algebraic expressions by making them easier to handle. The goal is to express a fraction in its simplest form by dividing the numerator and the denominator by their greatest common factor. In our exercise, we have the fraction \(\frac{-6 \pm 3\sqrt{3}}{3}\). The denominator is 3, so we'll divide each term in the numerator by this number:

• Divide \(-6\) by 3 to get \(-2\).
• Divide \(3\sqrt{3}\) by 3 to get \(\sqrt{3}\).

Now, the expression \(\frac{-6 \pm 3\sqrt{3}}{3}\) reduces to \(-2 \pm \sqrt{3}\). This shows the importance of checking each part of the numerator for simplification opportunities by the denominator.

Numerator and Denominator Simplification

Simplifying expressions involves attention to both numerators and denominators. After simplifying the square root and reducing the fraction, each component plays a role:

• Ensure each term in the numerator is reduced individually by dividing by the common factor from the denominator.
• All parts of the numerator interact independently with the denominator, as shown by \(\frac{-6}{3} \pm \frac{3\sqrt{3}}{3}\).

This methodical breakdown ensures all aspects of the expression are simplified properly, leading to the final form \(-2 \pm \sqrt{3}\). This reduction clarifies whether any further simplification is necessary or possible.