Problem 13 - 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. $$ \frac{\sqrt{27 b}}{\sqrt{6 b^{2}}} $$

Step-by-Step Solution

Answer

The simplest form of the quotient is \( \frac{3 \sqrt{2}}{2b} \) after rationalizing the denominator.

Step 1: Simplify the Radicals

We begin by simplifying the radicals in both the numerator and the denominator. The numerator is \( \sqrt{27b} \), which can be simplified because \( 27 = 9 \times 3 \). Thus, \( \sqrt{27b} = \sqrt{9 \times 3b} = \sqrt{9} \times \sqrt{3b} = 3\sqrt{3b} \). In the denominator, \( \sqrt{6b^2} = \sqrt{6} \times \sqrt{b^2} = \sqrt{6} \times b \) since \( \sqrt{b^2} = b \). Now we have: \[ \frac{3\sqrt{3b}}{b \sqrt{6}} \]

Step 2: Simplify the Fraction

Next, we simplify the fraction \( \frac{3\sqrt{3b}}{b \sqrt{6}} \). We notice that we can treat \( \sqrt{3b} \div \sqrt{6} \) by combining under a single radical as:\[ \sqrt{\frac{3b}{6}} = \sqrt{\frac{b}{2}} \]Now our expression becomes:\[ \frac{3}{b} \times \sqrt{\frac{b}{2}} \]

Step 3: Simplify Further and Write in Simplest Form

Since \( \sqrt{\frac{b}{2}} = \frac{\sqrt{b}}{\sqrt{2}} \), we can write the expression as:\[ \frac{3 \times \sqrt{b}}{b \times \sqrt{2}} = \frac{3 \sqrt{b}}{b \sqrt{2}} \]To simplify further, cancel \( b \) from \( \sqrt{b} \) giving us:\[ \frac{3 \sqrt{b}}{b \sqrt{2}} = \frac{3 \sqrt{b}}{b \sqrt{2}} \left(b eq 0\right) \]

Step 4: Combine Terms

Finally, simplify \( \frac{3 \sqrt{b}}{b \sqrt{2}} \) so we are left with only the coefficients, radicals, and eliminate unnecessary \( b \):\[ \frac{3}{\sqrt{2b}} \] Then rationalize the denominator by multiplying both the numerator and denominator by \( \sqrt{2} \) to give:\[ \frac{3 \sqrt{2}}{2b} \]

Key Concepts

Radicals

A radical is simply another way to express a root, such as a square root or cube root. You might have seen the radical sign, which looks like this: \( \sqrt{} \). It's like a magical portal that gives us the quantity that multiplies by itself to reach the number inside the radical. For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).

Sometimes, radicals involve variables, like \( \sqrt{b} \). We simplify radicals by attempting to take as much out of the 'inside' of the radical as possible. This helps make the expression more elegant and tidy. Whenever a number like 27 is inside a radical, we factor it into perfect squares. So, \( \sqrt{27} \) becomes \( 3 \sqrt{3} \) because 9 (a perfect square) times 3 gives us 27. The perfect square (9) comes out of the radical as its square root, which is 3.

This simplification process makes subsequent steps, like dealing with fractions or more complex expressions, much easier.

Rationalizing Denominators

The process of rationalizing denominators addresses a simple goal: eliminate radicals from the bottom (or denominator) of a fraction. This step is crucial because having radicals as denominators is often frowned upon in mathematics due to historical preferences or for simplification purposes.

To rationalize the denominator, we multiply both the top and bottom of the fraction by a radical that will cancel out the radical in the denominator. Let's say we have \( \frac{1}{\sqrt{2}} \). By multiplying top and bottom by \( \sqrt{2} \), we get \( \frac{\sqrt{2}}{2} \) instead. This cleans up the fraction by removing any radicals from the denominator.

• Identify the radical in the denominator.
• Multiply the fraction by a form of 1 that uses that radical, like \( \frac{\sqrt{2}}{\sqrt{2}} \).
• Simplify the new expression.

This process ensures clarity and keeps equations nice and uniform.

Fraction Simplification

Simplifying fractions is like tidying up a messy room; it involves reducing fractions to their simplest form. This might mean canceling common factors in the numerator and the denominator or, in the case of radical fractions, simplifying the radical parts too.

Take a fraction with radicals, like \( \frac{3 \sqrt{b}}{b \sqrt{2}} \). If \( b eq 0 \), the tactic is to see if any parts of the numerator and the denominator can "cancel" each other out. In this equation, the \( b \) terms in the numerator and denominator cancel each other, leaving us with \( \frac{3 \sqrt{b}}{\sqrt{2}} \).

• Check if there's a common factor in numbers or variables you can cancel.
• Simplify any radicals involved, if possible.
• Keep the neatest form to avoid clutter.

As with cleaning, the trick is in the details. Carefully handling each part makes the whole expression simpler and much more manageable.