Problem 16 - Algebra 2 and Trigonometry

Question

In \(3-29\) write each quotient in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \frac{\sqrt{12 a^{2}}}{\sqrt{4 a}} $$

Step-by-Step Solution

Answer

The quotient \( \frac{\sqrt{12a^2}}{\sqrt{4a}} \) simplifies to \( \sqrt{3a} \).

Step 1: Simplify the Radicals

First, simplify the radicals in the numerator and denominator. The expression is \( \frac{\sqrt{12a^2}}{\sqrt{4a}} \). By simplifying, \( \sqrt{12a^2} = \sqrt{12} \times \sqrt{a^2} = \sqrt{4 \times 3} \times a = 2a\sqrt{3} \).Similarly, \( \sqrt{4a} = \sqrt{4} \times \sqrt{a} = 2\sqrt{a} \).

Step 2: Write the Quotient of Radicals

Now, substitute the simplified radicals back into the expression:\(\frac{2a\sqrt{3}}{2\sqrt{a}}.\)

Step 3: Simplify the Fraction

Cancel out the common factor of 2 in the numerator and the denominator:\( \frac{2a\sqrt{3}}{2\sqrt{a}} = \frac{a\sqrt{3}}{\sqrt{a}} \).Next, simplify by canceling \( a \) and using the property \( \sqrt{a} = a^{1/2} \):\( \frac{a^{1} \times 3^{1/2}}{a^{1/2}} = a^{1-1/2}\sqrt{3} = a^{1/2}\sqrt{3} \).

Step 4: Final Simplified Form

Combine the expressions into the simplest form.The simplified form is \( \sqrt{3a} \) since \( a^{1/2} = \sqrt{a} \).

Key Concepts

Properties of Radicals

Understanding the properties of radicals is essential for simplifying radical expressions effectively. Radicals, most commonly square roots, adhere to specific rules that help in their simplification and manipulation. One of the key properties is the product rule for radicals, which states that \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \). This rule allows us to break down complex expressions into simpler, more manageable components.

• Another important property is the quotient rule: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), assuming \( b eq 0 \). This property is fundamental when dealing with fractions underneath the square root sign.
• Finally, knowing that \( \sqrt{a^2} = a \) when \( a \) is non-negative, empowers us to simplify expressions where variables are involved, such as \( \sqrt{12a^2} \).

Applying these properties, we can simplify radical expressions and make complex problems more intuitive and easier to solve.

Radical Expressions

A radical expression is any mathematical expression containing a radical symbol \( \sqrt{} \). These expressions can become quite elaborate, but understanding their components helps in simplifying them. In our original exercise, the radical expression is \( \frac{\sqrt{12a^2}}{\sqrt{4a}} \). This consists of two separate radicals, one in the numerator and one in the denominator.

By applying the properties of radicals, we first simplify each part. For example, \( \sqrt{12a^2} \) can be broken down into \( 2a\sqrt{3} \) by factoring 12 into 4 and 3, and recognizing \( \sqrt{a^2} = a \). Similarly, \( \sqrt{4a} \) becomes \( 2\sqrt{a} \).

• These simplifications make it easier to work with the fractions in radical expressions.
• Understanding how to manipulate these expressions aids in performing operations like addition, subtraction, and most importantly division, as seen in our example.

The more comfortable you become with these expressions, the better equipped you will be to tackle more complex algebraic problems.

Fraction Simplification

Simplifying fractions is a foundational skill in algebra that makes handling expressions involving fractions less daunting. In this exercise, simplifying the radical expression involves reducing the fraction \( \frac{2a\sqrt{3}}{2\sqrt{a}} \).

The first simplification step is to cancel common factors in the numerator and denominator - here, the factor of 2. This leaves us with \( \frac{a\sqrt{3}}{\sqrt{a}} \). Canceling further, using the property \( \sqrt{a} = a^{1/2} \), you divide the powers of \( a \) as follows: \( a^{1} / a^{1/2} = a^{1 - 1/2} = a^{1/2} \), which itself is \( \sqrt{a} \).

• Combining these steps, the simplified form becomes \( \sqrt{3a} \).
• This example demonstrates the virtue of carefully breaking down and recomposing fractions for simplicity and accuracy.

Recognizing patterns and applying radical and fraction simplification rules allows for more strategic and error-free problem solving.