Problem 5 - 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. $$ \sqrt{72} \div \sqrt{8} $$

Step-by-Step Solution

Answer

The simplest form is 3.

Step 1: Rewriting the Problem

Rewriting the problem using the given expression: \( \frac{\sqrt{72}}{\sqrt{8}} \).

Step 2: Using Properties of Radicals

Apply the property of radicals \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \) to rewrite the expression as \( \sqrt{\frac{72}{8}} \).

Step 3: Simplifying the Fraction Under the Radical

Simplify the fraction under the radical, \( \frac{72}{8} = 9 \). Thus, the expression becomes \( \sqrt{9} \).

Step 4: Calculating the Square Root

Calculate the square root of 9, which is 3. So, \( \sqrt{9} = 3 \).

Key Concepts

Quotient of Square Roots

When dealing with the quotient of square roots, you are simply dividing one square root by another. Let's explore this with the example of \( \sqrt{72} \div \sqrt{8} \).

• The first step is to express the division of two square roots as a fraction: \( \frac{\sqrt{72}}{\sqrt{8}} \).
• Instead of dealing with each square root separately, use the property of radicals, which allows you to combine them into a single square root: \( \sqrt{\frac{72}{8}} \). This property is quite handy as it simplifies the computation significantly.
• Think of it like this: rather than finding two square roots and then dividing, you're simplifying the process by first consolidating the fraction.
• This makes complex calculations easier and helps keep your work neat and organized.

Understanding and applying this principle effectively can save you time and avoid unnecessary complications.

Properties of Radicals

Radicals can be both fascinating and intricate. Using their properties correctly can simplify many mathematical tasks, such as dealing with square roots. Here's an overview of key properties aiding in simplification:

• **Product Property**: This states \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). It allows you to multiply numbers under separate radicals to combine them under one radical, simplifying your calculations.
• **Quotient Property**: As applied in the square roots quotient, the property \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \) enables you to transform a division of radicals into a single radical. This property made it easier to work with \( \frac{\sqrt{72}}{\sqrt{8}} \) as \( \sqrt{\frac{72}{8}} \).
• **Simplification Property**: If a number under the radical is a perfect square, such as \( \sqrt{9} \), simplify it to its integer form (3 in this case).

These properties streamline operations with radicals, clearing the path for solving what might seem at first glance to be a challenging problem. They bolster comprehension by providing structured ways to tackle expressions involving radicals, ensuring accurate and efficient solutions.

Simplifying Fractions

Simplifying fractions is a fundamental skill that plays a crucial role when working with radicals. Let's look at how this skill fits into our example:

• After rewriting the radicals using the property \( \sqrt{\frac{a}{b}} \), the core of the simplification is inside \( \frac{72}{8} \).
• Simplifying \( \frac{72}{8} \) is as easy as dividing the numerator and the denominator by their greatest common divisor (GCD), which is 8 here. Doing this gives \( 9 \).
• This reduced form is now much easier to manage within the square root operation.
• Recognizing when numbers can be simplified this way is key to mastering more complex algebraic problems.

By always reducing fractions to their simplest form, you ensure your calculations are as straightforward and error-free as possible. This integrated approach of working through fractions and radicals simultaneously enhances your mathematical proficiency and confidence.