Problem 6 - 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. $$ \sqrt{50 a^{3}} \div \sqrt{5 a} $$

Step-by-Step Solution

Answer

The quotient simplifies to \( a\sqrt{10} \).

Step 1: Write the Expression as a Fraction

First, express the division of the square roots as a single fraction: \( \frac{\sqrt{50a^3}}{\sqrt{5a}} \).

Step 2: Apply the Quotient Rule for Radicals

Use the property \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \) to combine the square roots: \( \sqrt{\frac{50a^3}{5a}} \).

Step 3: Simplify the Fraction Inside the Radical

Simplify the fraction inside the square root by dividing the numerator and the denominator by \(5a\): \( \frac{50a^3}{5a} = \frac{50}{5} \cdot \frac{a^3}{a} = 10a^2 \).

Step 4: Simplify the Square Root

Now simplify \( \sqrt{10a^2} \). Recall \( \sqrt{a^2} = a \) when \(a\) is non-negative. The expression becomes \( \sqrt{10} \cdot a \).

Step 5: Write the Final Simplified Form

The simplified form of the quotient \( \sqrt{50a^3} \div \sqrt{5a} \) is \( a\sqrt{10} \).

Key Concepts

Quotient Rule for Radicals

The quotient rule for radicals is an essential tool for simplifying expressions with square roots, especially when you are working with division. This rule states that the square root of a quotient is equal to the quotient of the square roots. Mathematically, this can be expressed as \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). To apply this rule, simply combine the radicands (the numbers under the radicals) into a single square root, allowing for simpler calculations afterward.

For instance, dividing \( \sqrt{50a^3} \) by \( \sqrt{5a} \) results in a single square root, \( \sqrt{\frac{50a^3}{5a}} \). This step is crucial as it reduces the complexity of the expression, enabling further simplification.

Keep this rule handy whenever you encounter division in radical expressions—it makes the process much more manageable and direct.

Radical Expressions

Radical expressions include any expressions containing a radical symbol, such as a square root. Understanding how to manipulate these expressions is crucial for simplifying them effectively. When working with radicals, especially square roots, there are several rules and properties that can assist.

• The product rule allows \( \sqrt{ab} = \sqrt{a} \times \sqrt{b} \).
• The quotient rule as discussed is \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \).

Knowing these properties helps break radical expressions into more manageable parts, enabling further simplification if necessary. In the exercise \( \sqrt{50a^3} \), start by simplifying the radicand. Recognize that \( 50 = 5 \times 10 \) and that powers like \( a^3 \) can be broken down when under a square root. These insights provide a valuable foundation for simplification.

Simplification Process

The simplification process involves several steps that reduce a complex expression to its simplest form. In radical expressions, this process often involves rationalization, combining radicals, and reducing numbers and variables within the radical sign.

For example, consider \( \sqrt{\frac{50a^3}{5a}} \). Simplify the fraction inside the square root by canceling common factors in the numerator and denominator. This involves dividing each term by \(5a\): \( \frac{50a^3}{5a} = 10a^2 \). Now, you have a simple expression under the square root, which is easier to work with.

Completing the simplification, take \( \sqrt{10a^2} \). Apply the rule \( \sqrt{a^2} = a \) to separate the square of \(a\) from the rest, resulting in \( a\sqrt{10} \). This final expression is both neat and easy to understand, showing how each part of the initial equation has been addressed and simplified.

Fraction Simplification

Fraction simplification is a fundamental skill necessary for working with rational expressions, especially those including radicals. This process involves reducing the numerator and the denominator to their simplest form by eliminating common factors.

In the solution \( \frac{50a^3}{5a} \), the goal is to divide both the top and bottom by their greatest common divisor, \( 5a \). This yields \( 10a^2 \), removing unnecessary terms and simplifying the expression greatly.

The ability to simplify fractions, especially within more complex mathematical operations, enhances understanding and reduces potential errors, making complex problems seem straightforward. This skill is critical when rerouting expressions through steps like the quotient rule for radicals, as it ensures you're always working with the simplest version of each component.