Question

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{-\sqrt{3}}{\sqrt{8}}$$

Short answer

The rationalized form of \(\frac{-\text{sqrt}(3)}{\text{sqrt}(8)}\) is \(\frac{-\text{sqrt}(6)}{4}\).

Step-by-step solution

Identify the expression

The given expression is \(\frac{-\text{sqrt}(3)}{\text{sqrt}(8)}\). The goal is to rationalize the denominator.

Simplify the denominator

First, note that \(\text{sqrt}(8) = \text{sqrt}(4 \times 2) = \text{sqrt}(4) \times \text{sqrt}(2) = 2\text{sqrt}(2)\).

Multiply by the conjugate

To rationalize the denominator, multiply both the numerator and the denominator by \(\text{sqrt}(2)\): \(\frac{-\text{sqrt}(3)}{2\text{sqrt}(2)} \times \frac{\text{sqrt}(2)}{\text{sqrt}(2)} = \frac{-\text{sqrt}(3) \times \text{sqrt}(2)}{2\text{sqrt}(2) \times \text{sqrt}(2)}\).

Simplify the expression

This will result in: \(\frac{-\text{sqrt}(6)}{2 \times 2} = \frac{-\text{sqrt}(6)}{4}\).

Key concepts

Radicals

In mathematics, a radical is often referred to as a root, such as a square root or cube root. Radicals are expressed with the radical symbol (√). For example, the square root of 9 is written as \( \sqrt{9} \), which simplifies to 3.

When dealing with radicals in algebra, especially when simplifying expressions, it helps to break down the number or expression inside the radical as much as possible. Factor the number into its prime components to see if you can simplify the radical further. For instance, \( \sqrt{8} \) can be broken down into \( \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2} \).

Understanding how to handle radicals is crucial for rationalizing denominators, a common algebra task.

Square Roots

A square root of a number is a value that, when multiplied by itself, equals the original number. The square root symbol is \( \sqrt{} \), and if we apply it to a number like 25, we get \( \sqrt{25} = 5 \) because \( 5 \times 5 = 25 \).

For expressions that involve square roots, aim to simplify them. If the number inside the square root is a product, see if any factors are perfect squares. For example:
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3} \]
For rationalization, it’s key to make the denominator free of square roots, often by multiplying by an appropriate form of 1. In our example, we turned \( \sqrt{8} \) into \( 2 \sqrt{2} \) and used \( \sqrt{2} \) to eliminate the root.

Simplifying Expressions

Simplifying expressions refers to making an algebraic expression as simple as possible. This involves combining like terms, reducing fractions, and eliminating radicals in the denominator. Simplifying expressions often makes solving equations easier.

Let's look at the example given in the exercise. We started with \( \frac{-\sqrt{3}}{\sqrt{8}} \). First, we simplified the denominator \( \sqrt{8} \) to \( 2 \sqrt{2} \).

We then rationalized the denominator by multiplying both the numerator and the denominator by \( \sqrt{2} \). This gave us \( \frac{-\sqrt{3} \times \sqrt{2}}{2\sqrt{2} \times \sqrt{2}} = \frac{-\sqrt{6}}{4} \). Simplifying expressions like this often include key steps such as combining, factoring, and reducing terms.

Algebra

Algebra introduces us to the world of variables, constants, and operators (such as addition and multiplication). This branch of math uses symbols to represent numbers in equations and expressions.

In the given problem, the goal was to rationalize the denominator of the expression \( \frac{-\sqrt{3}}{\sqrt{8}} \). This is a classic algebra exercise where manipulation of radicals helps achieve a simplified form.

Algebra provides the tools to:

• Manipulate equations
• Understand relationships between variables
• Simplify complex expressions

By mastering algebraic techniques, you can solve a variety of mathematical problems, including rationalizing denominators efficiently.