Problem 49 - Algebra A Combined Function

Question

Rationalize each denominator and simplify if possible. See Section 10.5. $$ \frac{1}{\sqrt{3}} $$

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{\sqrt{3}}{3} \).

Step 1: Identify the Problem

The given expression is \( \frac{1}{\sqrt{3}} \). The denominator contains a square root, which we need to rationalize.

Step 2: Multiply by the Conjugate

To rationalize the denominator, multiply both the numerator and the denominator by \( \sqrt{3} \), the conjugate of \( \sqrt{3} \). This gives us: \[ \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{3} \times \sqrt{3}} \]

Step 3: Simplify the Denominator

The denominator becomes \( \sqrt{3} \times \sqrt{3} = 3 \) since \( \sqrt{x} \times \sqrt{x} = x \). Thus, the expression simplifies to: \[ \frac{\sqrt{3}}{3} \]

Step 4: Final Simplified Expression

The expression \( \frac{\sqrt{3}}{3} \) is the simplified form with the denominator rationalized. There are no further simplifications possible in this expression.

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental concept in algebra that involves making a math problem easier to understand and solve. It usually means reducing the number of terms or making an expression more straightforward.
One way to simplify is to combine like terms or eliminate unnecessary parts in a fraction. For instance, when you multiply or divide by the same value in the numerator and the denominator, you technically change the form of the expression without altering the value.
In our example, simplifying occurs after rationalizing the denominator; instead of leaving the expression as \( \frac{\sqrt{3}}{3} \), we made sure each part was reduced to a more manageable state.

Square Roots in Denominators

Having a square root in the denominator of a fraction is something we typically aim to avoid in mathematics. This practice is called rationalizing the denominator. It helps in creating an equivalent fraction that is easier to interpret or further use in calculations.
The main trick in dealing with square roots in denominators is to multiply both the numerator and the denominator by a specific term that will eliminate the square root. If the denominator is \( \sqrt{3} \), like in our original problem, you multiply the entire fraction by \( \frac{\sqrt{3}}{\sqrt{3}} \).
This manipulation doesn't change the value of the fraction because you are essentially multiplying by 1, but it will remove the square root from the denominator, presenting a cleaner expression.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both contain an algebraic expression. Dealing with these kinds of fractions can involve several unique algebraic techniques to simplify or rationalize the expression, as necessary.
Understanding algebraic fractions is essential because they appear frequently in equations and real-world problem-solving scenarios. To properly simplify them, you often need to apply a combination of arithmetic and algebraic rules, such as factoring or employing the distributive property.
The solution of our exercise illustrates a typical approach when faced with a radical in an algebraic fraction’s denominator.

• We create an equivalent fraction by rationalizing the denominator, ensuring it no longer contains radicals.
• This form is usually easier to work with in further equations or mathematical applications.

With these basic principles, handling algebraic fractions becomes much more manageable.