Problem 3 - 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. $$ \sqrt{24} \div \sqrt{6} $$

Step-by-Step Solution

Answer

The simplest form is 2.

Step 1: Use the Quotient Property of Square Roots

The Quotient Property of Square Roots says \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \) if \( b eq 0 \). Applying this property to the expression gives us \( \sqrt{\frac{24}{6}} \).

Step 2: Simplify Inside the Square Root

Simplify the fraction inside the square root: \( \frac{24}{6} = 4 \). So the expression becomes \( \sqrt{4} \).

Step 3: Simplify the Square Root

Calculate the square root of 4. Since \( \sqrt{4} = 2 \), the simplest form of the expression is 2.

Key Concepts

Quotient Property of Square Roots

The quotient property of square roots is a powerful tool in algebra that allows us to simplify radical expressions when dividing. The rule is expressed as \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \), provided \( b eq 0 \). This means that when you have two square root terms being divided, you can combine them into a single square root over a fraction.

• Why do we need this? It simplifies the process of dealing with complex radical expressions, making calculations more straightforward.
• When using this property, always start by ensuring that the expression in the denominator is not zero, as division by zero is undefined.

In the given exercise, the expression \( \sqrt{24} \div \sqrt{6} \) becomes \( \sqrt{\frac{24}{6}} \) using this property. This sets the stage for further simplification within the square root.

Simplifying Expressions

Simplifying expressions is a key step in making complex mathematical operations easier to handle. Once an expression is in a simpler form, it becomes much easier to evaluate or use in further calculations.

• This process often involves reducing fractions and breaking down complex terms.
• In the context of the original problem, after using the quotient property, we simplify \( \frac{24}{6} \) to obtain the number 4.

The fraction \( \frac{24}{6} \) simplifies directly to 4 because 24 is a multiple of 6. This simplification is critical because it transforms the problem from a complex radical division into a simple square root, \( \sqrt{4} \), making the final steps of the calculation much more straightforward.

Mathematical Operations with Radicals

Radicals, and specifically square roots, are common in many areas of mathematics and require specific techniques for manipulation and simplification. Operations involving radicals can seem daunting but following systematic methods helps in managing them efficiently.

• Essential operations include addition, subtraction, multiplication, and division of radicals.
• Each operation has its own set of rules, similar to those of regular integers, but radicals must be like (same radicand) to be added or subtracted.
• In division, as seen in the original exercise, the quotient property allows for simplification and is particularly useful.

After simplifying the fraction inside the radical expression, \( \sqrt{4} \) is easy to solve since we know \( \sqrt{4} = 2 \). This demonstrates the effectiveness of combining mathematical operations with a systematic approach, ensuring every step works toward reducing the complexity of the problem.