Problem 49 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{\sqrt{5}}{\sqrt{12}}\)

Step-by-Step Solution

Answer

\( \frac{\sqrt{5}}{\sqrt{12}} = \frac{\sqrt{15}}{6} \).

Step 1: Simplify the denominator

To simplify \( \frac{\sqrt{5}}{\sqrt{12}} \), we need to simplify \( \sqrt{12} \). The number 12 can be factored into \( 4 \times 3 \), where 4 is a perfect square. So, \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \).

Step 2: Simplify the fraction

Now substitute \( \sqrt{12} \) with \( 2\sqrt{3} \): \( \frac{\sqrt{5}}{2\sqrt{3}} \). This is a simplified version of the original expression.

Step 3: Rationalize the denominator

Multiply the numerator and the denominator by \( \sqrt{3} \) to eliminate the radical in the denominator: \[ \frac{\sqrt{5}}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{15}}{2 \times 3} = \frac{\sqrt{15}}{6}. \]

Step 4: Verify the simplest form

Check the simplified expression, \( \frac{\sqrt{15}}{6} \), to ensure there are no common factors or radicals that can be further simplified. Since 15 has no perfect square factors other than 1, \( \sqrt{15} \) is in its simplest form.

Key Concepts

Rationalizing Denominators

Rationalizing denominators is a technique used to make the denominator of a fraction a rational number by eliminating any radical expressions it contains. For instance, when faced with an expression like \( \frac{\sqrt{5}}{2\sqrt{3}} \), the goal is to remove the radical from the denominator. To do this, we multiply both the numerator and the denominator by the radical that is in the denominator.

• First, identify the radical in the denominator. In this case, it is \( \sqrt{3} \).
• Multiply the numerator and the denominator by this radical: \( \frac{\sqrt{5}}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \).
• This results in \( \frac{\sqrt{15}}{6} \), where the denominator is now rational.

By rationalizing the denominator, you ensure the expression is in a more conventional and standard form, making it easier to work with in further calculations or comparisons.

Simplest Radical Form

The simplest radical form of an expression means that the number under the radical sign is expressed in the most reduced form possible, without any perfect squares (other than 1) as factors. In simpler terms, you want to ensure the value inside the square root is minimized.

To find the simplest radical form, take for example \( \sqrt{12} \):

• Begin by factoring the number under the square root. Factor 12 as \( 4 \times 3 \), where 4 is a perfect square.
• Express the square root of the perfect square separately: \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} \).
• Simplify further by calculating \( \sqrt{4} = 2 \), resulting in \( 2\sqrt{3} \).

Now, \( 2\sqrt{3} \) is in its simplest radical form. Simplifying radicals like this makes subsequent computations and comparisons more straightforward.

Factoring Under a Square Root

Factoring under a square root involves breaking down the number inside the radical into its prime factors or as a product of integers, aiming to identify any perfect square factors. Simplifying a radical in this way helps in reducing it to its simplest radical form.

Consider \( \sqrt{12} \) as an example:

• First, factor 12 into its components: \( 12 = 4 \times 3 \).
• Recognize that 4 is a perfect square, making it possible to simplify: \( \sqrt{12} = \sqrt{4 \times 3} \).
• Separate the square root of the perfect square: \( \sqrt{4} = 2 \), so \( \sqrt{12} = 2\sqrt{3} \).

Factoring under a square root simplifies radicals by allowing you to "take out" a perfect square from under the root, resulting in expressions that are often easier to understand and work with in algebraic operations.