Problem 9 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{12}{\sqrt{27}}\)

Step-by-Step Solution

Answer

The simplified form of the fraction is \( \frac{4\sqrt{3}}{3} \).

Step 1: Identify the Denominator

The denominator of the given fraction is \( \sqrt{27} \). Our task is to rationalize it, which means we want to get rid of the square root in the denominator so that it becomes a rational number.

Step 2: Simplify the Square Root

Simplify \( \sqrt{27} \) by expressing 27 as a product of perfect squares. We know that 27 = 9 x 3, so \( \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3} \). Thus, \( \frac{12}{\sqrt{27}} = \frac{12}{3\sqrt{3}} \).

Step 3: Rationalize the Denominator

To remove the radical, multiply both the numerator and the denominator by \( \sqrt{3} \). This gives \( \frac{12}{3\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \), simplifying to \( \frac{12 \sqrt{3}}{3 \times 3} = \frac{12 \sqrt{3}}{9} \).

Step 4: Simplify the Fraction

Now, simplify \( \frac{12 \sqrt{3}}{9} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. Therefore, \( \frac{12 \sqrt{3}}{9} = \frac{12 \sqrt{3} \div 3}{9 \div 3} = \frac{4 \sqrt{3}}{3} \).

Key Concepts

Simplifying Square Roots

Simplifying square roots is all about making the square root expression simpler, preferably into integers or products of integers and square roots without complicating radicals. To do this, you need to find the factors of the number inside the square root and see if any of them are perfect squares.

For example, when you have \(\sqrt{27}\), you can break down 27 into 9 and 3 because 9 is a perfect square. This simplifies \(\sqrt{27}\) into \(\sqrt{9} \times \sqrt{3} = 3\sqrt{3}\).

This simplification reduces the complexity from a larger non-perfect square into a more understandable form. This step not only aids in mathematics but also helps make further calculations easier.

Fractions

Fractions represent parts of a whole, written in the form \(\frac{a}{b}\) where 'a' is the numerator and 'b' is the denominator. Working with fractions often involves operations like simplification, addition, subtraction, multiplication, and division.

When dealing with fractions that have radicals in the denominator, like \(\frac{12}{\sqrt{27}}\), it becomes important to rationalize. Rationalizing means changing the fraction so that there are no square roots or radicals in the denominator, turning it into a simpler form that is widely accepted in mathematical writing.

Simplest Form

Writing a fraction in its simplest form involves ensuring that the numerator and the denominator have no common factors other than 1. This means reducing or simplifying the fraction as much as possible.

For example, \(\frac{12 \sqrt{3}}{9}\) simplifies to its simplest form, \(\frac{4 \sqrt{3}}{3}\), by dividing both the numerator and the denominator by their greatest common divisor, which is 3 in this case.

This process ensures that the fraction is as simple as possible, making it easier to work with and understand for further calculations. The simplest form is ideal as it represents the smallest equivalent version of the fraction.

Perfect Squares

A perfect square is an integer that can be expressed as the square of another integer. For instance, numbers like 4, 9, 16, and 25 are perfect squares because they result from squaring 2, 3, 4, and 5, respectively.

In algebra, recognizing perfect squares can significantly simplify radical expressions. When simplifying a square root, such as \(\sqrt{27}\), breaking it down into factors, such as \(\sqrt{9 \times 3}\), helps simplify it to \(3\sqrt{3}\). The factor 9 is a perfect square, critical in reducing the complexity of the square root.

Understanding perfect squares facilitates easier manipulation of square roots, especially when rationalizing fractions, ensuring calculations are precise and manageable.