Problem 39 - College Algebra

Question

In Exercises \(39-48\), rationalize the denominator. $$\frac{1}{\sqrt{7}}$$

Step-by-Step Solution

Answer

\(\frac{\sqrt{7}}{7}\)

Step 1: Identify the denominator

The denominator of the given fraction is \( \sqrt{7} \). So, we aim to rationalize \( \sqrt{7} \).

Step 2: Rationalize the denominator

To rationalize the denominator, we can multiply both the numerator and denominator by \( \sqrt{7} \). Doing this, we get: \( \frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{7}}{7} \).

Key Concepts

Fraction Simplification

Fraction simplification is an important process in mathematics. It involves reducing a fraction to its simplest form. This means there are no common factors left between the numerator and the denominator except 1. Simplifying fractions can make them easier to understand and compare.
To simplify, you often divide both the numerator and the denominator by their greatest common divisor. For instance, if you have a fraction like \( \frac{8}{12} \), you divide both by 4, the greatest common factor, to get \( \frac{2}{3} \).
When dealing with fractions that involve roots, part of simplification may include rationalizing the denominator so it becomes a rational number. This often makes further simplification or computation easier.

Square Roots

Square roots are numbers which, when multiplied by themselves, yield the original number. For example, the square root of 16 is 4 because \(4 \times 4 = 16\). Square roots are often represented with the radical symbol \(\sqrt{}\).
Square roots are irrational in many cases, like \(\sqrt{2}\) or \(\sqrt{3}\), meaning they cannot be expressed as a simple fraction. However, square roots have important properties and uses in mathematics.
For example, knowing that \(\sqrt{7} \times \sqrt{7} = 7\) lets us rationalize denominators in a fraction, which removes square roots from the denominator by converting it into a whole number. This is a crucial part of making fractions more manageable.

Rational Numbers

Rational numbers can be expressed as the quotient of two integers where the denominator is not zero, like \(\frac{3}{4}\) or 5. They include whole numbers, finite decimals, and repeating decimals.
Rationalizing involves converting an expression into a form where the denominator is a rational number. For example, in the fraction \(\frac{1}{\sqrt{7}}\), the denominator \(\sqrt{7}\) is irrational.
To rationalize this, we multiply both the numerator and the denominator by \(\sqrt{7}\), giving us \(\frac{\sqrt{7}}{7}\). Now, the denominator is a rational number, making the fraction easier to use in calculations. Understanding rational numbers helps simplify complex expressions and solve various mathematical problems.