Problem 42 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{\frac{8}{25}}\)

Step-by-Step Solution

Answer

\(\frac{2\sqrt{2}}{5}\)

Step 1: Simplify the Fraction

The square root of a fraction \(\sqrt{\frac{a}{b}}\) is the same as the square root of \(a\) divided by the square root of \(b\). Therefore, we can write \(\sqrt{\frac{8}{25}}\) as \(\frac{\sqrt{8}}{\sqrt{25}}\).

Step 2: Simplify the Denominator

Find the square root of the denominator. Here, \(\sqrt{25} = 5\). Therefore, the fraction becomes \(\frac{\sqrt{8}}{5}\).

Step 3: Simplify the Numerator

Next, simplify \(\sqrt{8}\). The number 8 can be factored into 4 and 2, where 4 is a perfect square. Thus, \(\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}\).

Step 4: Write in Simplest Radical Form

Replace \(\sqrt{8}\) with its simplified form in the fraction. This gives us \(\frac{2\sqrt{2}}{5}\). Therefore, the simplest radical form of \(\sqrt{\frac{8}{25}}\) is \(\frac{2\sqrt{2}}{5}\).

Key Concepts

Simplest Radical Form

The simplest radical form is the most reduced version of a radical expression, where no perfect square factors other than 1 remain in the radicand (the number inside the square root). This process involves simplifying the expression by extracting perfect squares from the radicand.

• To find the simplest radical form, break down the number inside the radical (the radicand) into its prime factors.
• Identify and extract any perfect squares.
• Multiply the extracted perfect squares together and place them outside the radical.

For example, when simplifying \(\sqrt{8}\), identify that 8 can be expressed as 4 and 2. Since 4 is a perfect square, it is extracted, resulting in 2 outside the radical, leaving you with \(2\sqrt{2}\). When applied to a fraction, this process must be applied separately to the numerator and the denominator.

Radicals

Radicals are expressions that include roots, most commonly square roots. Solving radical expressions involves both understanding their manipulative properties and following set rules for simplification to make them easier to work with or to solve equations.

• Radicals are helpful in solving problems that involve squares and roots, where we need to "undo" the power of 2 process.
• The radical sign (\(\sqrt{}\)) represents the root operation, which can be square roots, cube roots, etc.

When working with radicals in fractions, they can be split into the numerator and denominator for easier simplification. For instance, \(\sqrt{\frac{8}{25}}\) can be expressed as \(\frac{\sqrt{8}}{\sqrt{25}}\), allowing for separate simplification of each part by finding their simplest radical forms. Remember, the presence of perfect squares is key in simplifying any radical expression effectively.

Square Roots

Square roots are a type of radical expression where we try to find a number that, when multiplied by itself, gives the original number under the radical.

• The operation is denoted with the radical symbol and without an index when dealing with square roots, as they are the most common.
• Perfect squares, like 4, 9, 16, and 25, are numbers whose square roots are whole numbers.

A classic example is \(\sqrt{25}\), which equals 5 because 5 times 5 equals 25. Recognizing perfect squares when simplifying expressions like \(\frac{\sqrt{8}}{\sqrt{25}}\) can aid immensely in reducing the expression to its simplest radical form. Always seek out the largest perfect square factor within a radicand to efficiently simplify the expression, resulting in the simplest form possible.