Problem 321 - Elementary Algebra

Question

In the following exercises, simplify. $$ \frac{2-\sqrt{32}}{8} $$

Step-by-Step Solution

Answer

\frac{1}{4} - \frac{\text{\text{√}\text{2}}}{2}}

Step 1: Simplify the square root

First, simplify the square root term in the numerator. Recognize that \(\text{32} = 16 \times 2\), so \(\text{\text{√}32} = \text{\text{√}(16 \times 2)} = \text{\text{√}16} \times \text{\text{√}2} = 4\text{\text{√}2}\).

Step 2: Substitute back into the expression

Replace \( \text{\text{√}\text{\text{32}}} \) with \( 4 \text{\text{√}\text{\text{2}}} \) in the original expression: \(\frac{2-4 \text{\text{√}\text{\text{2}}}}{8} \).

Step 3: Separate the fraction

Rewrite the fraction as two separate fractions: \(\frac{2}{8} - \frac{4 \text{\text{√}\text{\text{2}}}}{8} \).

Step 4: Simplify each fraction

Simplify each term in the fractions: \(\frac{2}{8} = \frac{1}{4} \) and \(\frac{4 \text{\text{√}\text{2}}}{8} = \frac{\text{\text{√}\text{2}}}{2} \). So, the expression becomes \( \frac{1}{4} - \frac{\text{\text{√}\text{2}}}{2} \).

Key Concepts

Square Roots

When simplifying expressions, understanding square roots is essential. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 times 4 equals 16. In algebra, square roots often appear under radical symbols, like \(\sqrt{32}\).

To simplify square roots, factorize the number inside the radical to its prime factors or into perfect squares. For instance, \(32\) can be written as \(16 \times 2\). By recognizing that 16 is a perfect square (\(4^2\)), we can simplify \(\sqrt{32}\) to 4\sqrt{2}. This approach can help break down complex square roots into simpler, more manageable terms.

Fractions

Handling fractions is a fundamental skill in algebra. Fractions represent parts of a whole and consist of a numerator (top part) and a denominator (bottom part). In the exercise provided, fractions are split and each part is simplified individually.

For example, the expression \(\frac{2-\sqrt{32}}{8}\) is divided into \(\frac{2}{8}\) and \(\frac{4\sqrt{2}}{8}\). This step helps to simplify each fraction separately:

\(\frac{2}{8} = \frac{1}{4}\)
\(\frac{4\sqrt{2}}{8} = \frac{\sqrt{2}}{2}\).

This method makes it easier to simplify expressions involving fractions by breaking them down.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. This process makes equations easier to understand and solve. To simplify algebraic expressions, follow these steps:

Identify and simplify square roots.
Break down complex fractions into simpler parts.
Combine like terms and reduce fractions to their lowest terms.

In the given example, the original expression \(\frac{2-\sqrt{32}}{8}\) is simplified by

Simplifying \(\sqrt{32}\) to \(4\sqrt{2}\).
Breaking the fraction into \(\frac{2}{8}\) and \(\frac{4\sqrt{2}}{8}\).
Reducing to \(\frac{1}{4} - \frac{\sqrt{2}}{2}\).

Following these strategies systematically helps in simplifying algebraic expressions accurately.