Problem 22 - 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. $$ \cdot \sqrt{\frac{a^{3}}{5 b}} $$

Step-by-Step Solution

Answer

\(\frac{a \sqrt{5ab}}{5b}\)

Step 1: Simplify the numerator of the radical

The numerator of the expression under the square root is \(a^3\). Recall the rule \(\sqrt{x^2} = x\) for simplifying a square root. Therefore, you can write \(a^3\) as \(a^2 \cdot a\). Apply the square root to \(a^2\), \(\sqrt{a^2} = a\), and leave the \(a\) inside the square root.

Step 2: Simplify the denominator of the radical

The denominator under the square root is \(5b\). Since neither term under the root for 5 or \(b\) is a perfect square, they remain within the square root.

Step 3: Write the simplified expression

Following steps 1 and 2, the square root of \(\frac{a^3}{5b}\) can be simplified by taking \(a\) out of the square root (since \(a^3\) was written as \(a^2 \cdot a\)). The expression simplifies to \(\frac{a}{\sqrt{5b}} \cdot \sqrt{a}\).

Step 4: Rationalize the denominator

To ensure a simplified radical form, the denominator should not have a square root. Multiply both the numerator and the denominator by \(\sqrt{5b}\). This results in the numerator becoming \(a \cdot \sqrt{a} \cdot \sqrt{5b}\) and the denominator simplifying to \(5b\).

Step 5: Final Simplified Radical Form

Combine the numerator terms under one square root to obtain \(a \sqrt{5ab}\) as final result since multiplying two radicals results in the product of the quantities under the root. The simplified radical form is \(\frac{a \sqrt{5ab}}{5b}\).

Key Concepts

Radical Expressions

Radical expressions feature a radical sign, which is typically a square root, cube root, or higher-level root. In this exercise, we focused on the square root, symbolized as \(\sqrt{}\). These expressions allow us to manage and simplify complex numbers without the need for decimals.

The expression \(\sqrt{\frac{a^{3}}{5b}}\) involves both a numerator and a denominator under the radical sign. Simplifying such expressions requires careful steps. Look at the numerator and the denominator independently:

• For the numerator \(a^3\), breaking it into \(a^2 \cdot a\) allows for easy root extraction, since \(\sqrt{a^2} = a\).
• The denominator, \(5b\), does not present any simple squares. Thus, it remains unchanged under the radical.

Ultimately, breaking down each part individually leads us closer to its simplified radical form. The final expression can often look more elegant and is easier to work with in further calculations.

Rationalizing Denominators

Rationalizing a denominator involves ensuring no radical remains in the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by a suitable radical that cancels out the one in the denominator. In our example, we had \(\frac{a}{\sqrt{5b}}\).

To rationalize it:

• Multiply both the numerator and the denominator by \(\sqrt{5b}\).
• The denominator then becomes \( 5b \), as \(\sqrt{5b} \cdot \sqrt{5b} = 5b\).
• The numerator becomes \(a \cdot \sqrt{a} \cdot \sqrt{5b}\), which after simplifying gives \(a \sqrt{5ab}\).

This clears the radical from the denominator, giving us a more simplistic and usable expression \(\frac{a\sqrt{5ab}}{5b}\). Rationalizing denominators ensures the expression looks neater and adheres to standard mathematical presentation rules.

Radical Form

The term "radical form" refers to an expression that involves a root symbol. In mathematics, keeping expressions in radical form is standard practice as it retains more information and provides precise values.

In our exercise, the goal is finding the simplest radical form of \(\sqrt{\frac{a^{3}}{5b}}\). This involved:

• Breaking down each component under the root.
• Simplifying where possible and rationalizing to prevent radicals in the denominator.

The simplest radical form was \(\frac{a\sqrt{5ab}}{5b}\). By following steps for simplification and rationalization, expressions retain their logical and elegant structure, which is valuable when solving complex math problems. Whether dealing with integers or variables, maintaining radical form is key for clarity and efficiency.

Square Roots

Square roots specifically deal with finding a number that, when multiplied by itself, gives the original value under the radical sign. This concept allows computation of radicals in more digestible forms.

For instance, the exercise provided \(\sqrt{a^3}\). By expressing \(a^3\) as \(a^2 \cdot a\), the square root allows us to extract \(a\) out, leaving \(\sqrt{a}\) behind under the radical:

• \(\sqrt{a^2} = a\)
• Remaining under the root is \(\sqrt{a}\)

Square roots afford a way to manage powers and simplify them into expressions that are more practical for further manipulation. It's particularly useful in algebraic contexts to achieve a more refined expression outcome.