Problem 4 - 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. $$ \sqrt{75} \div \sqrt{3} $$

Step-by-Step Solution

Answer

The simplest form is 5.

Step 1: Understand the Problem

We have the expression \(\sqrt{75} \div \sqrt{3}\). Both terms are square roots, and we need to divide one by the other. We can use the property of square roots that allows us to simplify this expression.

Step 2: Apply the Quotient Rule for Square Roots

The quotient rule for square roots states that \(\sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}}\). Use this rule to rewrite the expression: \(\sqrt{75} \div \sqrt{3} = \sqrt{\frac{75}{3}}\).

Step 3: Simplify the Fraction Inside the Square Root

Simplify the fraction \(\frac{75}{3}\). By dividing 75 by 3, we get 25: \(\frac{75}{3} = 25\). Thus, our expression now is: \(\sqrt{25}\).

Step 4: Simplify the Square Root

The square root of 25 is a well-known value, since \(25 = 5^2\). Thus, \(\sqrt{25} = 5\).

Key Concepts

Quotient Rule for Square Roots

When dealing with the division of two square roots, there's a wonderful property called the quotient rule for square roots. This rule greatly simplifies the process. It tells us that instead of dividing one root by another, we can combine them under a single square root. Mathematically, if you have \(\sqrt{a} \div \sqrt{b}\), the rule allows you to rewrite this as \(\sqrt{\frac{a}{b}}\). This transformation eases the task of simplifying expressions.
Using this property means fewer steps and errors when working through square root problems. Instead of splitting up each number under its own root, you put them together, making the problem much simpler and less cluttered.

Properties of Square Roots

Square roots have unique properties that are incredibly handy when simplifying expressions. Here are a few key properties:

• Non-negative Outputs: The square root function outputs only non-negative values, even if the radicand (the number under the root) is positive.
• Product Property: The square root of a product is the product of the square roots. That is, \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).
• Quotient Property: As we've seen, \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\) is a very useful tool when dividing square roots.

These properties help us navigate through complex expressions with ease and allow for the flexible combination and separation of terms under square roots.

Simplifying Fractions

Simplifying fractions is an essential skill, especially when you encounter them inside square roots. The fraction \(\frac{a}{b}\) is simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).
Let's do a quick rundown using our exercise as an example: starting with \(\frac{75}{3}\), we find the GCD is 3. By dividing both 75 and 3 by their GCD, we get \(\frac{75}{3} = 25\).
This simplification transforms the expression into something much easier to work with. By maintaining this approach, you can effectively handle fractions within different mathematical contexts, particularly those that involve square roots.