Problem 12 - Algebra 1

Question

Simplify the expression. $$\frac{6}{\sqrt{10}}$$

Step-by-Step Solution

Answer

\( \frac{3 * \sqrt{10}}{5} \)

Step 1: Identify and Rationalize the Denominator

The denominator contains a square root. To rationalize the denominator, multiply both the numerator and denominator of the fraction by the square root found in the denominator. In this case, multiply by \( \sqrt{10} \). This results in \( \frac{6*\sqrt{10}}{\sqrt{10}*\sqrt{10}} \).

Step 2: Carry out the Multiplication

In the result of Step 1, carry out the multiplication operation. This results in \( \frac{6*\sqrt{10}}{10} \).

Step 3: Simplify the Fraction

In the result of Step 2, see if you can simplify the fraction. For \( \frac{6*\sqrt{10}}{10} \), both the numerator and denominator are divisible by 2. So, we can simplify the fraction to \( \frac{3 * \sqrt{10}}{5} \).

Step 4: Write the Final Answer

The final answer is \( \frac{3 * \sqrt{10}}{5} \), which is the fraction in simplest form with a rationalized denominator.

Key Concepts

Rationalizing the Denominator

When you have a fraction with a square root in the denominator, it can look a bit messy. That's why we try to "rationalize" it, or in simple terms, make it a bit friendlier. The goal is to get rid of that pesky square root at the bottom. To do this, you multiply both the top and the bottom of the fraction by the same square root that's in the denominator.
For example, with \( \frac{6}{\sqrt{10}} \), our denominator is \( \sqrt{10} \). So, we'll multiply both the numerator and the denominator by \( \sqrt{10} \), giving us \( \frac{6\sqrt{10}}{\sqrt{10}\sqrt{10}} \).

This step is critical because multiplying the denominator's square root with itself, results in a whole number. In this case, \( \sqrt{10} \times \sqrt{10} = 10 \). Now our fraction looks a lot simpler.

Fraction Simplification

Once you have rationalized the denominator, the next step is to simplify the fraction. This means making it as easy to read as possible by reducing the fraction. Look at both the numerator and the denominator to see if they have any common factors.
Let's take \( \frac{6\sqrt{10}}{10} \) from our last step. The numbers 6 and 10 both have a common factor of 2. So, divide them both by 2.

Doing the division gives you \( \frac{3\sqrt{10}}{5} \). We've now simplified the fraction, making it cleaner and easier to handle in any further calculations.

Square Roots

Square roots are numbers that give you a particular value when multiplied by themselves. For instance, the square root of 10 is written as \( \sqrt{10} \). When you multiply \( \sqrt{10} \times \sqrt{10} \), you get back 10.

It's essential to understand square roots because they often appear in fractions needing simplification. They can seem tricky at first, but remember that the aim is usually to use properties of these square roots to make expressions simpler.

• In most rationalizing tasks, you'll multiply the square root with itself, turning it into a regular whole number.
• Practice recognizing common square roots, which can help speed up your problem-solving.

Square roots are found everywhere, from geometry to algebra, making them an indispensable part of mathematics.