Problem 11 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The simplified form of the expression is \( \frac{3\sqrt{10}}{5} \)

Step 1: Identify the irrational number

First, identify the irrational number in the denominator. In this case, the irrational number is \( \sqrt{10} \).

Step 2: Rationalize the denominator

In order to rationalize the denominator, multiply both the numerator and the denominator by the irrational number \( \sqrt{10} \). This gives us \( \frac{6 \cdot \sqrt{10}}{\sqrt{10} \cdot \sqrt{10}} \). The denominator now becomes 10 since \( \sqrt{10} \cdot \sqrt{10} = 10 \)

Step 3: Simplify the fraction

The new fraction is \( \frac{6\sqrt{10}}{10} \). We can simplify this fraction by dividing both the numerator and the denominator by 2. This gives us \( \frac{3\sqrt{10}}{5} \)

Key Concepts

Simplifying Radical Expressions

A radical expression involves roots, such as square roots or cube roots. Simplifying these expressions means making them easier to work with by eliminating any radicals that can be reduced or simplified. The main goal is to make the expression less complex without changing its value. This often involves:

• Reducing the expression under the radical sign as much as possible.
• Combining radicals that are alike.
• Rationalizing the denominator if there's a radical present there.

In our example, the expression \( \frac{6}{\sqrt{10}} \) involves a square root in the denominator. To simplify this, we multiply by a form of 1 that contains the radical in question (in this case, \( \sqrt{10} \)) in both the numerator and the denominator. This step eliminates the radical in the denominator, simplifying it to a whole number and making it easier to work with.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both are algebraic expressions. These can include variables, constants, or even radicals like in our given expression \( \frac{6}{\sqrt{10}} \). Working with algebraic fractions typically involves:

• Ensuring that the fractions are in simplest form.
• Rationalizing the denominator, especially if it contains radicals.
• Finding a common denominator when adding or subtracting algebraic fractions.

Let's look at the example: when we rationalize the denominator by multiplying by \( \sqrt{10}/\sqrt{10} \), the expression changes to \( \frac{6\sqrt{10}}{10} \). After simplifying further by dividing both the numerator and the denominator by 2, we obtain the simplest form: \( \frac{3\sqrt{10}}{5} \). This is an excellent example of simplifying an algebraic fraction.

Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a simple fraction. Their decimal representation goes on forever without repeating. A common example is a square root of a non-perfect square, such as \( \sqrt{10} \). These values pose unique challenges in mathematics, especially when they appear in the denominator of a fraction.
When an irrational number is in the denominator, the fraction is not in the simplest form and needs rationalizing. This means converting the fraction so its denominator is a rational number, typically achieved by multiplying the numerator and denominator by an appropriate value, as we did in the above example.
Understanding how to work with irrational numbers is crucial for students as they often appear in various mathematical problems and calculations, particularly in geometry and algebra.