Problem 14 - 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. $$ \frac{3}{\sqrt{3} x} $$

Step-by-Step Solution

Answer

The simplest form is \( \frac{\sqrt{3}}{x} \).

Step 1: Simplify the Radical in the Denominator

The given expression is \( \frac{3}{\sqrt{3} x} \). The square root in the denominator makes it difficult to work with, so our first step is to remove it. We multiply both the numerator and the denominator by \( \sqrt{3} \) to rationalize the denominator. This gives us: \[ \frac{3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3} x} = \frac{3 \sqrt{3}}{3x} \]

Step 2: Perform Cancellation

Now we have \( \frac{3 \sqrt{3}}{3x} \). Notice that there is a common factor of \( 3 \) in the numerator and the denominator. We cancel this common factor to simplify the expression further: \[ \frac{3 \sqrt{3}}{3x} = \frac{\sqrt{3}}{x} \]

Step 3: Final Result

We have simplified the initial quotient by rationalizing the denominator and performing cancellation. The simplest form of the given expression is \( \frac{\sqrt{3}}{x} \).

Key Concepts

Rationalizing the Denominator

When you come across a fraction with a denominator involving a radical, it is often easier to work with once you remove the radical. This process is known as rationalizing the denominator. In the expression \( \frac{3}{\sqrt{3} x} \), the denominator contains \( \sqrt{3} \). To rationalize it, you need to eliminate the square root from the denominator.
To do this, multiply both the numerator and the denominator by \( \sqrt{3} \). This is a clever trick because multiplying by \( \sqrt{3} \) essentially leaves the value of the expression unchanged, but transforms the denominator into a rational number rather than an irrational one:
- Original: \( \frac{3}{\sqrt{3} x} \)- Multiply by \( \sqrt{3} \): \( \frac{3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3} x} \)- Result: \( \frac{3 \sqrt{3}}{3x} \)
Now the denominator is \( 3x \), and the expression is much simpler to work with.

Simplifying Radicals

Radicals often look complicated, but simplifying them can make expressions much cleaner and easier to handle. Let's break down how to do it.
The square root \( \sqrt{3} \) can typically be left in its current form because 3 is a prime number, which means there are no other factors to simplify it further. However, when radicals appear in a denominator, they can be multiplied out to simplify the fraction, as demonstrated in the rationalizing step.
To simplify radicals, including ones that can be broken down into the product of simpler factors, follow these steps:

• Look for perfect squares under the radical sign, like \( \sqrt{4}, \sqrt{9}, \sqrt{16} \), etc.
• Extract these perfect square factors from the radical.
• Simplify what's left under the radical as much as possible.

This process leads to cleaner mathematical expressions and makes further algebraic manipulation more straightforward.

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but they include variables along with numbers. Simplifying them follows familiar principles: finding common factors and reducing the fraction to its simplest form.
In the expression \( \frac{3 \sqrt{3}}{3x} \), this means identifying common factors of 3 in both the numerator and the denominator, which allows for cancellation. Cancelling out these common factors results in the simplified form \( \frac{\sqrt{3}}{x} \).
Some tips for simplifying algebraic fractions include:

• Look for common numerical factors in the numerator and denominator.
• If variables are present, check for common variable factors that can be canceled out.
• Remember that only terms that are multiplied (not added or subtracted) can be cancelled.

This makes algebraic fractions easier to work with, whether you're solving equations or simplifying expressions further.