Problem 18 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{\frac{4 a^{4}}{25}} $$

Step-by-Step Solution

Answer

The simplest radical form of \( \sqrt{\frac{4 a^4}{25}} \) is \( \frac{2a^2}{5} \).

Step 1: Separating the Radical

The given expression is \( \sqrt{\frac{4a^4}{25}} \). We can use the property of square roots to separate it into two parts: \( \sqrt{\frac{4a^4}{25}} = \frac{\sqrt{4a^4}}{\sqrt{25}} \).

Step 2: Simplifying the Denominator

The denominator of our expression is \( \sqrt{25} \), which simplifies to \( 5 \) because \( \sqrt{25} = 5 \). So our expression becomes \( \frac{\sqrt{4a^4}}{5} \).

Step 3: Simplifying the Numerator

Now, let's simplify the numerator \( \sqrt{4a^4} \). This can be separated and simplified as \( \sqrt{4} \times \sqrt{a^4} \). We know that \( \sqrt{4} = 2 \), and \( \sqrt{a^4} = a^2 \), assuming \( a \) is non-negative. Thus, \( \sqrt{4a^4} = 2a^2 \).

Step 4: Combining Simplified Components

With the numerator simplified to \( 2a^2 \), we substitute back into our fraction expression: \( \frac{2a^2}{5} \).

Step 5: Final Result

The simplest radical form of \( \sqrt{\frac{4 a^4}{25}} \) is \( \frac{2a^2}{5} \).

Key Concepts

Square Root Properties

When working with square roots, there are a few properties that make simplifying expressions much easier. The property of taking the square root of a fraction is especially helpful. It tells us that

• \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)

This property allows you to separate the radicals of the numerator and the denominator, simplifying each independently.
Another important property is the ability to separate a product under a square root. This means you can write

• \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)

This feature is useful when the radicand (the number inside the radical) can be factored into perfect squares, making it easier to simplify.Knowing that the square root of a perfect square will yield a whole number significantly reduces the complexity of our problem. For instance, \( \sqrt{4} = 2 \) or \( \sqrt{25} = 5 \) because these are perfect squares.

Simplifying Fractions

When we have a fraction that includes square roots, simplifying it requires careful manipulation. The goal is to make these fractions as clear and simple as possible. Begin by ensuring that the denominator is simplified down to a simple whole number, if possible. For example, in \( \frac{\sqrt{4a^4}}{\sqrt{25}} \), we know that \( \sqrt{25} = 5 \). Thus, the expression can be reduced to \( \frac{\sqrt{4a^4}}{5} \).
Next, focus on the numerator. If it contains a perfect square or can be factored into perfect squares, separate and simplify these. By writing products under a single square root separately, you can deal with each part independently. Here, \( \sqrt{4a^4} \) can be simplified by recognizing that both \( 4 \) and \( a^4 \) are perfect squares, leading to \( 2a^2 \).
Finally, put everything back together to achieve the simplest form of your original fraction. If all parts are simplified, the expression will be straightforward and easy to understand, like in our final result of \( \frac{2a^2}{5} \).

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that helps in reducing complex expressions to simpler forms. This process often involves breaking down a given expression into parts that are easy to handle, such as taking apart a square root of a fraction and simplifying independently.

• Separate and simplify square roots using known properties.
• Simplify fractions wherever possible by reducing numerator and denominator to their simplest forms.
• Recombine the simplified components to create an easier and more comprehensible expression.

In the case of \( \sqrt{\frac{4a^4}{25}} \), these steps are applied. Once you have separated and simplified each part, the simple recombination of \( 2a^2 \) over \( 5 \) leads to the final simple and elegant form \( \frac{2a^2}{5} \).
By following the processes of separation, simplification, and recombination, any daunting algebraic expression becomes less intimidating and much more manageable.