Problem 60 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{-6 \sqrt{5}}{\sqrt{18}}\)

Step-by-Step Solution

Answer

The expression simplifies to \(-\sqrt{10}\).

Step 1: Simplify the Denominator

The expression given is \( \frac{-6 \sqrt{5}}{\sqrt{18}} \). We start by simplifying \( \sqrt{18} \). Since 18 = 9 \times 2, we can rewrite this as \( \sqrt{9 \times 2} \), which simplifies to \( \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \). Thus, the expression becomes \( \frac{-6 \sqrt{5}}{3\sqrt{2}} \).

Step 2: Simplify the Fraction

Now, simplify the fraction \( \frac{-6 \sqrt{5}}{3\sqrt{2}} \). Divide both the numerator and the denominator by the common factor 3: \( \frac{-6}{3} = -2 \), so the expression simplifies to \( \frac{-2 \sqrt{5}}{\sqrt{2}} \).

Step 3: Rationalize the Denominator

To eliminate the square root from the denominator, multiply both the numerator and the denominator by \( \sqrt{2} \): \( \frac{-2 \sqrt{5} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{-2 \sqrt{10}}{2} \).

Step 4: Simplify the Resulting Expression

Simplify \( \frac{-2 \sqrt{10}}{2} \) by dividing both the numerator and the denominator by 2, resulting in \( -\sqrt{10} \). Since both 2's cancel out, the simplest radical form of the expression is \( -\sqrt{10} \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a key concept in mathematical simplification that involves removing any radical expressions from the denominator of a fraction. This process ensures the expression is in its simplest form, making it easier to interpret and work with in further calculations.
To rationalize a denominator, we aim to eliminate any radicals, such as square roots, that may appear there. In our example, let's consider the fraction \( \frac{-2\sqrt{5}}{\sqrt{2}} \). The square root in the denominator can complicate calculations, so we remove it by multiplying both the numerator and the denominator by the radical itself.
This means performing the operation \( \frac{-2\sqrt{5}\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}} \). The denominator \( \sqrt{2} \times \sqrt{2} \) simplifies to 2, because the multiplication of a square root by itself results in the original number inside the root.
The numerator becomes \( -2\sqrt{10} \), and thus the fraction simplifies to \( \frac{-2\sqrt{10}}{2} \). Ultimately, this method effectively transfers any radical component from the bottom of a fraction, thus streamlining the overall expression.

Simplifying Radicals

Simplifying radicals involves reducing the expression within a root sign to its most basic form. The goal is to simplify numerical values or variables under the square root to create a more manageable expression. In our example, we begin with \( \sqrt{18} \), which is part of the denominator in the initial expression.
Recognizing that 18 can be factored into \( 9 \times 2 \), we take the square root of each factor: \( \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} \). Since \( \sqrt{9} = 3 \), we write the expression as \( 3\sqrt{2} \).
By doing this, the radical component is reduced to its simplest form. Similarly, in problems involving variables, breaking down the factors inside the radical makes it easier to simplify and solve the entire expression.

• Identify factors that are perfect squares.
• Simplify these factors outside the radical sign.
• Ensure all like terms are expressed in the simplest radical form.

With this approach, radicals become less intimidating, allowing for clearer mathematical operations.

Concept of Square Roots

The concept of square roots is foundational in understanding how to manage and manipulate radical expressions. A square root asks what number, when multiplied by itself, equals the original number beneath the radical sign. For example, \( \sqrt{9} \) equals 3 because \( 3 \times 3 = 9 \).
Square roots are vital when simplifying expressions and solving equations. They help break down numbers or expressions to their simplest form, which is essential in various areas of math and science. Remember, the square root of a positive number has two potential values: one positive and one negative, indicated by \( \pm \).
In our exercise, we used the property of square roots to simplify \( \sqrt{18} \) into \( 3\times\sqrt{2} \). By understanding and leveraging the properties of square roots, we simplify expressions containing radicals efficiently.
When dealing with square roots:

• Identify perfect square factors within the radical.
• Remember that square roots will always yield two principal roots: positive and negative.
• Apply knowledge of square roots to rationalize and simplify other expressions effectively.

This basic yet powerful concept allows for the thorough simplification of complex mathematical expressions.