Problem 20 - Algebra 1: Concepts and Skills

Question

Determine whether the radical expression is in simplest form. Explain. $$ \sqrt{\frac{2}{8}} $$

Step-by-Step Solution

Answer

No, the radical expression \(\sqrt{\frac{2}{8}}\) was not in its simplest form. In its simplest form, the expression is \(\frac{1}{2}\) .

Step 1: Simplify the expression under the square root

Simplify the fraction \(\frac{2}{8}\) to its simplest form, which is \(\frac{1}{4}\).

Step 2: Apply the square root to the simplified expression

Apply the square root to \(\frac{1}{4}\). The square root of \(\frac{1}{4}\) is \(\frac{1}{2}\).

Step 3: Compare the result to the original expression

Comparing the result, \(\frac{1}{2}\), to the original expression, \(\sqrt{\frac{2}{8}}\), it can be seen that the original expression was not in its simplest form.

Key Concepts

Radical Expression Simplification

When we talk about simplifying radical expressions, we're referring to the process of making the expression as straightforward as possible. This often involves removing any perfect squares from under the radical, reducing fractions, and eliminating any radical in the denominator of a fraction.

A key step in this process is breaking down the number inside the radical—referred to as the radicand—into its prime factors and seeing if any of those factors are perfect squares. If they are, they can be taken out from under the radical, simplifying the expression. For instance, \( \sqrt{8} \) simplifies to \( \sqrt{4\times 2} \) and further to \( 2\sqrt{2} \) since 4 is a perfect square. It's a way of streamlining the expression for easier calculation and understanding.

Square Root Operations

Performing square root operations is about understanding how to process and simplify the square root of a number or expression. The square root of a number is a value that, when multiplied by itself, gives the original number. When an expression under a square root, like a fraction, cannot be simplified any further, you have reached the simplest radical form.

For example, taking the square root of a simple fraction involves finding the square root of both the numerator and the denominator separately. With \( \sqrt{\frac{1}{4}} \) as in our original exercise, since both 1 and 4 are perfect squares, you can simply calculate their square roots: \( \sqrt{1} = 1 \) and \( \sqrt{4} = 2 \) resulting in \( \sqrt{\frac{1}{4}} = \frac{1}{2} \). When working with radicals, patience and attention to detail can prevent mistakes and ensure the simplification is correct.

Fraction Simplification

Fraction simplification is an essential skill in mathematics that helps to make problem-solving easier. Simplifying a fraction means expressing it in its most reduced form, where the numerator and the denominator have no common factors other than one. There are two main ways to simplify a fraction: by finding the greatest common divisor (GCD) of the numerator and denominator or by dividing both by common factors.

In the exercise, the fraction \( \frac{2}{8} \) was reduced to \( \frac{1}{4} \) by recognizing that both 2 and 8 are divisible by 2. By doing so, we ensure that the radical expression we are analyzing is in its lowest terms, which makes the subsequent square root operation much simpler. Simplifying fractions not only makes the numbers more manageable but also helps in identifying whether the original expression is already in its simplest form.