Problem 1 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{2}}{\sqrt{7}}\)

Step-by-Step Solution

Answer

The rationalized form is \(\frac{\sqrt{14}}{7}\).

Step 1: Understand the problem

We need to rationalize the denominator of the fraction \(\frac{\sqrt{2}}{\sqrt{7}}\). Rationalizing the denominator involves removing any square roots or irrational numbers from the denominator of the fraction.

Step 2: Multiply by the Conjugate

To rationalize \(\frac{\sqrt{2}}{\sqrt{7}}\), multiply both the numerator and the denominator by \(\sqrt{7}\). This will remove the square root in the denominator because \( \sqrt{7} \times \sqrt{7} = 7\).

Step 3: Perform the multiplication

Carry out the multiplication: \[\frac{\sqrt{2} \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}} = \frac{\sqrt{14}}{7}\].The denominator \(\sqrt{7} \times \sqrt{7}\) simplifies to 7.

Step 4: Simplify the Expression

Now, we have \(\frac{\sqrt{14}}{7}\). Since there are no further simplifications possible, this is our rationalized expression.

Key Concepts

Radicals

A radical, often expressed with the square root symbol \(\sqrt{}\), is a way to indicate the root of a number. In mathematics, we primarily encounter square roots, but radicals can represent other roots as well, such as cube roots or fourth roots.
In the expression \(\frac{\sqrt{2}}{\sqrt{7}}\), the \(\sqrt{2}\) and \(\sqrt{7}\) are radicals. The purpose of using radicals is to represent numbers that are not perfect squares. When working with them, it's essential to understand that the square root of a number \(x\) is a number that, when multiplied by itself, gives \(x\).

• Radicals are used to simplify expressions that involve irrational numbers.
• The goal is to express them in their simplest form which sometimes involves removing the radical from the denominator.
• Understanding radicals is crucial for rationalizing denominators and solving various algebraic expressions.

This concept is particularly vital in problems where you need to rationalize denominators, as it helps you determine the appropriate way to multiply and manipulate these expressions.

Simplifying Expressions

Simplifying expressions is a core algebraic skill which involves rewriting expressions in as compact and simple a form as possible.
In this context, simplifying the expression \(\frac{\sqrt{2}}{\sqrt{7}}\) involves removing the square roots from the denominator by rationalizing it.
This makes the resulting expression easier to work with and understand. Simplification typically includes:

• Combining like terms in expressions.
• Reducing fractions to their simplest form.
• Eliminating unnecessary radicals from denominators by multiplying appropriately.

After rationalizing and simplifying \(\frac{\sqrt{2} \times \sqrt{7}}{7}\), we arrive at the expression \(\frac{\sqrt{14}}{7}\). Here, \(\sqrt{14}\) cannot be simplified further, and the expression is fully simplified. This makes it both easier to interpret and more useful in further calculations.

Multiplying Conjugates

The concept of multiplying conjugates is often used in rationalizing denominators. A conjugate in mathematics typically refers to a pair of expressions of the form \((a + b\sqrt{c})\) and \((a - b\sqrt{c})\).
However, in this particular exercise, we focus on using the same radical to eliminate the denominator, since it's not accompanied by a binomial term. When you multiply radicals like \(\sqrt{7}\) with \(\sqrt{7}\), you simplify the expression to a rational number, since \(\sqrt{7} \times \sqrt{7} = 7\).

• Multiplying conjugates removes radicals from denominators and simplifies expressions.
• This method ensures that the denominator is a rational number, which is more manageable and easier to interpret.
• It is an essential algebraic technique for simplifying expressions with radical denominators.

By using this method to multiply \(\sqrt{7}\) to both the numerator and the denominator, we've accomplished our goal of rationalizing \(\frac{\sqrt{2}}{\sqrt{7}}\) to \(\frac{\sqrt{14}}{7}\).