Problem 350 - Elementary Algebra

Question

In the following exercises, simplify and rationalize the denominator. $$ \frac{10}{\sqrt{11}} $$

Step-by-Step Solution

Answer

MULTIPWORK{{step{textbf{\textbf{\textbf{\textbf{{{{{{{{{{{{{{{{{{{{{{{NUMERATOR'''''''''''''''}}} \ MULTIPNUMERATOR Step numeratortextbf{\textbf{\textbf{\textbf{{{{{{{{√11}{textbf√}{sqrt{11}numerator =Mult THE SQUAREROOT

Step 1: Identify the Problem

We need to simplify and rationalize the given expression: \ \ \ \ \ \ \ \ $$ \frac{10}{\textcolor{red}{\boldsymbol{\bm{\bm{\bm{\bm{\bm{\bm{\bm{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{\textcolor{red}{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{\textcolor{red}{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{\textbf{{{{{{{{{{{√{√}{√}{√{√∫11}{sqrt{sqrt{sqrt}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}\(11}}$$11}}$$11}}\) which contains a square root in the denominator that we need to remove.

Step 2: Rationalizing the Denominator

To rationalize the denominator, we multiply both the numerator and the denominator by the square root found in the denominator, \ $$ \frac{10}{\textbf{\textbf{\textbf{\textbf{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{√}\textr{{{{11}{sqrt{11}}$$$$, so we multiply by ∗

Key Concepts

simplifying fractions

Simplifying fractions involves reducing them to their simplest form. This is an essential skill for solving many mathematical problems. A fraction is simplified when the numerator and the denominator have no common factors other than 1. In the exercise, we start with the fraction \(\frac{10}{\sqrt{11}}\). We need to simplify this by rationalizing the denominator. Simplifying keeps the fraction easy to understand and work with. To do it:

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and denominator by the GCF.

In our case, the numerator is 10, and the denominator is \(\sqrt{11}\). Let's now understand how square roots and rationalizing come into play.

square roots

Square roots are values that, when multiplied by themselves, give the original number. For instance, \(\sqrt{25} = 5\) since \(5 \times 5 = 25\).
In the given exercise, our denominator is \(\sqrt{11}\). Because square roots can sometimes complicate expressions, we often rationalize them. Why do we do this? An irrational number (one that cannot be expressed as a simple fraction) can make further calculations difficult. Consider a quick fact:

• \( \sqrt{a^2} = a\) for any non-negative number a.

Knowing how square roots function will help you rationalize the denominator efficiently.

denominator rationalization

Rationalizing the denominator means converting a fraction such that the denominator is a rational number. To do this when you encounter a square root in the denominator:

• Multiply both the numerator and the denominator by the same square root to eliminate it from the denominator.

This approach turns the denominator into a whole number. For our exercise:
\ \frac{10}{\sqrt{11}} \ should be multiplied by \( \frac{\sqrt{11}}{\sqrt{11}} \). This gives us:
\ \frac{10 \cdot \sqrt{11}}{\sqrt{11} \cdot \sqrt{11}} = \frac{10 \sqrt{11}}{11} \
So, \ \frac{10}{\sqrt{11}} = \frac{10 \sqrt{11}}{11} \
We have successfully rationalized the denominator and simplified the fraction. This process helps in performing further operations like addition, subtraction, and comparison effortlessly. Remember, practice makes perfect. Work on similar problems to get a good grasp on rationalizing denominators.