Problem 101 - College Algebra with Modeling and Visualization

Question

Rationalize the denominator. $$ \sqrt{\frac{b}{12}} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt{3b}}{6} \)

Step 1: Identify the Problem

We need to rationalize the denominator in the expression \( \sqrt{\frac{b}{12}} \). This means transforming the expression so there is no square root in the denominator.

Step 2: Rewrite as a Fraction Inside the Square Root

Start by expressing the square root of a fraction as the fraction of two square roots: \( \sqrt{\frac{b}{12}} = \frac{\sqrt{b}}{\sqrt{12}} \).

Step 3: Simplify the Denominator

Recognize that \( \sqrt{12} \) can be simplified. \( 12 = 4 \times 3 \), so \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4}\sqrt{3} = 2\sqrt{3} \). Thus, our expression becomes \( \frac{\sqrt{b}}{2\sqrt{3}} \).

Step 4: Rationalize the Denominator

Multiply both the numerator and the denominator by \( \sqrt{3} \) to eliminate the square root in the denominator: \[ \frac{\sqrt{b}}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{b} \cdot \sqrt{3}}{2 \cdot 3} = \frac{\sqrt{3b}}{6}. \]

Step 5: Final Expression

The expression \( \sqrt{\frac{b}{12}} \) rationalized is \( \frac{\sqrt{3b}}{6} \).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics often encountered in algebra and geometry. When we talk about the square root of a number, we refer to a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, since 3 \times 3 = 9. The square root symbol is denoted as \( \sqrt{\ } \).
It's important to know that not every number has a whole number as its square root. For instance, \( \sqrt{2} \) is an irrational number, meaning it cannot be expressed as a simple fraction.

• Square roots help in undoing the operation of squaring a number.
• If \( x^2 = y \), then \( x = \sqrt{y} \).

Understanding square roots is crucial when working with radical expressions, especially when simplifying or rationalizing denominators.

Simplifying Radicals

Simplifying radicals involves expressing a radical expression in its simplest form. This process often includes breaking down the number inside the square root into its prime factors. Consider the example from our problem, \( \sqrt{12} \).
Recognizing that 12 = 4 \times 3 allows us to separate it into \( \sqrt{4} \) and \( \sqrt{3} \). Since \( \sqrt{4} = 2 \), we can rewrite \( \sqrt{12} \) as \( 2\sqrt{3} \). The objective is to make calculations easier by reducing the numbers inside radicals.

• Simplification makes complex expressions manageable.
• Simplified radicals are clearer and less prone to error in calculations.

This step is vital before attempting to rationalize the denominator as it ensures the expression is as straightforward as possible.

Fractions in Algebra

Fractions are a key component of algebra. They represent parts of a whole and are written as a division of one number by another. When fractions appear within square roots, as in the problem \( \sqrt{\frac{b}{12}} \), they present unique challenges.
Rationalizing the denominator involves removing any radicals from the bottom of the fraction. This is done by introducing a term that will eliminate the radical when multiplied. In our example, multiplying by \( \sqrt{3} \) helped in achieving this, simplifying the expression from \( \frac{\sqrt{b}}{2\sqrt{3}} \) to \( \frac{\sqrt{3b}}{6} \).

• Fractions require understanding both numerators and denominators.
• Balancing both parts helps in simplifying algebraic expressions.

In algebra, the ability to manipulate fractions, especially those with radicals, is essential for solving equations and simplifying expressions efficiently.