Problem 345 - Elementary Algebra

Question

In the following exercises, simplify and rationalize the denominator. $$ \frac{10}{\sqrt{6}} $$

Step-by-Step Solution

Answer

\(\frac{5 \text{√}6}{3}\)

Step 1: Identify the Irrational Denominator

Recognize that the denominator of the fraction \(\frac{10}{\text{√}6}\) is irrational because it contains a square root.

Step 2: Multiply by the Conjugate

To rationalize the denominator, multiply the numerator and the denominator by the same square root of the denominator to make it rational. In this case, multiply both the numerator and denominator by \(\text{√}6\): \(\frac{10}{\text{√}6} \times \frac{\text{√}6}{\text{√}6} = \frac{10 \times \text{√}6}{\text{√}6 \times \text{√}6}\)

Step 3: Simplify the Denominator

Multiply the square roots in the denominator: \(\text{√}6 \times \text{√}6 = 6\), so now the fraction is: \(\frac{10 \text{√}6}{6}\)

Step 4: Simplify the Fraction

To simplify the fraction, divide both the numerator and the denominator by their greatest common divisor (GCD), which is 2 in this case: \(\frac{10 \text{√}6}{6} = \frac{10 \text{√}6 / 2}{6 / 2} = \frac{5 \text{√}6}{3}\)

Key Concepts

simplifying fractions

Simplifying fractions is an essential skill in mathematics because it makes calculations easier and results more understandable. When simplifying a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). This process reduces the fraction to its simplest form. For example, consider the fraction \(\frac{10 \text{√}6}{6}\). Here, both 10 and 6 are divisible by 2. Thus, we simplify it as follows:
\(\frac{10 \text{√}6}{6} = \frac{10 \text{√}6 / 2}{6 / 2} = \frac{5 \text{√}6}{3}\).
Simplifying fractions helps in rationalizing the denominator more efficiently by making the numbers involved smaller and easier to handle.

irrational numbers

Irrational numbers are numbers that cannot be written as simple fractions. They have non-repeating, non-terminating decimal parts. Common examples include \(\text{√}2\), \(\text{π}\), and \(\text{e}\). When you encounter an irrational number in the denominator of a fraction, you need to 'rationalize' it, meaning you convert the denominator to a rational number.

For instance, in the fraction \(\frac{10}{\text{√}6}\), the denominator \(\text{√}6\) is irrational. To rationalize it, we multiply both the numerator and the denominator by \(\text{√}6\):
\(\frac{10}{\text{√}6} \times \frac{\text{√}6}{\text{√}6} = \frac{10 \times \text{√}6}{\text{√}6 \times \text{√}6} = \frac{10 \text{√}6}{6}\).
This converts the denominator to a rational number, allowing for easier handling of the fraction.

square roots

Square roots are used to find a number that, when multiplied by itself, gives the original number. It’s denoted by the symbol \( \text{√} \). For instance, \( \text{√}4 = 2 \) because \( 2 \times 2 = 4 \). When dealing with fractions, square roots often appear and sometimes need special handling, especially if they are in the denominator.

In the exercise \(\frac{10}{\text{√}6}\), the denominator contains a square root, which is irrational. To simplify it, we employ the process of rationalization. We multiply both the numerator and the denominator by \(\text{√}6 \). Hence, \( \frac{10}{\text{√}6} \times \frac{\text{√}6}{\text{√}6} = \frac{10 \text{√}6}{6} \).
This removes the square root from the denominator, making the fraction easier to work with. Finally, simplifying the resulting fraction by dividing both the numerator and the denominator by their GCD results in \( \frac{5 \text{√}6}{3} \).