Problem 61 - Algebra and Trigonometry

Question

Rationalize the denominator. (a) \(\frac{1}{\sqrt{6}}\) (b) \(\frac{3}{\sqrt{2}}\) (c) \(\frac{9}{\sqrt{3}}\)

Step-by-Step Solution

Answer

(a) \(\frac{\sqrt{6}}{6}\), (b) \(\frac{3\sqrt{2}}{2}\), (c) \(3\sqrt{3}\)

Step 1: Understand the task

The task is to rationalize the denominator of a given fraction. Rationalizing the denominator involves removing any square roots present in the denominator by multiplying the numerator and denominator by a suitable expression.

Step 2a: Rationalize part (a) \(\frac{1}{\sqrt{6}}\)

To rationalize \(\frac{1}{\sqrt{6}}\), multiply both the numerator and denominator by \(\sqrt{6}\):\[ \frac{1}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{6}}{6} \]Now, the expression has a rational denominator.

Step 2b: Rationalize part (b) \(\frac{3}{\sqrt{2}}\)

To rationalize \(\frac{3}{\sqrt{2}}\), multiply both the numerator and denominator by \(\sqrt{2}\):\[ \frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2} \]This gives a rational denominator of 2.

Step 2c: Rationalize part (c) \(\frac{9}{\sqrt{3}}\)

To rationalize \(\frac{9}{\sqrt{3}}\), multiply both the numerator and denominator by \(\sqrt{3}\):\[ \frac{9}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{9\sqrt{3}}{3} \]Simplifying the fraction gives \(3\sqrt{3}\) as the result.

Key Concepts

Square Roots

A square root is a special number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 times 5 equals 25. Square roots are often denoted by the radical symbol \( \sqrt{} \).
When working with square roots, especially in mathematical expressions, the goal is often to simplify them. Simplifying might involve taking a number like \( \sqrt{8} \) and breaking it down. Since \( 8 = 4 \times 2 \), and \( \sqrt{4} = 2 \), it simplifies to \( 2\sqrt{2} \).
Remember:

• If a number is a perfect square (like 4, 9, 16), it will "come out" of the radical completely.
• If not, simplify as far as possible to make your calculations easier.

Fractions

Fractions are a way to express numbers that aren't whole. They consist of a numerator (the top part) and a denominator (the bottom part). For example, in the fraction \( \frac{1}{2} \), 1 is the numerator, and 2 is the denominator.
Fractions can represent parts of a whole or ratios. Operations with fractions often involve making them easier to read or work with, such as rationalizing the denominator. This means making the denominator a rational number, usually by removing any radicals.
When rationalizing a fraction's denominator:

• Multiply both the numerator and the denominator by the square root in the denominator to eliminate it.
• Remember that multiplying both parts of a fraction by the same thing doesn't change its value.

Simplifying Expressions

Simplifying expressions involves making them as simple as possible without changing their value. This process is key in mathematics to make calculations more manageable and to find equivalent expressions.
One aspect of simplifying is removing square roots from the denominator, known as rationalizing. After multiplying by the conjugate or the same square root, simply proceed to clean up the expression.
Steps to simplify:

• Factor where possible, to simplify square roots.
• Combine like terms if they exist.
• Ensure fractions are in their simplest form by dividing the numerator and denominator by their greatest common divisor if possible.

Applying these steps helps in making complex expressions easier to understand and work with.