Problem 94 - Intermediate Algebra

Question

Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the radical expression. See the Concept Check in this section. $$ \frac{5}{\sqrt{27}} $$

Step-by-Step Solution

Answer

Multiply both by \( \sqrt{3} \).

Step 1: Identify the radical denominator

In the fraction \( \frac{5}{\sqrt{27}} \), the denominator is \( \sqrt{27} \). We need to eliminate the square root in the denominator by rationalizing it.

Step 2: Simplify the expression under the radical

Start by simplifying \( \sqrt{27} \). Since 27 can be factored into prime numbers, \( 27 = 3^3 \). Therefore, \( \sqrt{27} = \sqrt{3^3} = 3\sqrt{3} \).

Step 3: Determine the factor needed for rationalization

To rationalize \( 3\sqrt{3} \), multiply both the numerator and the denominator by \( \sqrt{3} \). This will eliminate the square root in the denominator.

Step 4: Multiply and simplify the expression

Multiply the numerator and the denominator by \( \sqrt{3} \):\[\frac{5}{3\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3 \times 3} = \frac{5\sqrt{3}}{9}\]The denominator is now rationalized.

Step 5: Finalize the solution

The smallest number both the numerator and denominator should be multiplied by to rationalize the expression is \( \sqrt{3} \).

Key Concepts

Radical Expressions

Radical expressions involve mathematical expressions with a root symbol, like the square root (\(\sqrt{}\)). These roots can involve numbers or variables, making them an essential part of algebra. Understanding how to manage and manipulate these expressions is crucial when learning algebraic methods. Often, radicals in denominators need to be simplified or removed for precise mathematics. This is due to the preference for expressions to have integer or rational numbers as denominators. In our exercise, we deal with the expression \(\frac{5}{\sqrt{27}}\). The aim is to eliminate or rationalize the square root in the denominator, making the expression simpler and more standardized.

Simplifying Radicals

Simplifying radicals is the process of breaking down a radical expression into its simplest form. This involves reducing the number under the root to the smallest number possible. Let's take \(\sqrt{27}\) as an example.
First, recognize 27 is not a perfect square, meaning it cannot be simplified to an integer. So, we perform prime factorization: 27 is factored into \(3^3\). Therefore, \(\sqrt{27} = \sqrt{3^3}\).

Applying the property of radicals, where \(\sqrt{a^2 \cdot a} = a\sqrt{a}\), this simplifies to \(3\sqrt{3}\). The radical expression is now simpler, but its denominator still needs rationalization, which involves further operations.

Prime Factorization

Prime factorization is the method of expressing a number as the product of its prime numbers. This technique is helpful in simplifying radicals since it can reveal perfect squares under a radical that can be extracted.

In the exercise, prime factorization helps simplify \(\sqrt{27}\). We expressed 27 as \(3 \times 3 \times 3\) or \(3^3\). This reveals that \(3^2\) (a perfect square) can be taken out from under the square root, simplifying \(\sqrt{27}\) to \(3\sqrt{3}\).
Using prime factorization not only aids in simplifying radicals but also enhances the understanding of factor trees and repetitive prime multiplication inherent in numbers, making complex calculations more manageable.