Problem 118 - Elementary and Intermediate Algebra

Question

The following problems involve addition, subtraction, and multiplication of radical expressions, as well as rationalizing the denominator. Perform the operations and simplify, if possible. All variables represent positive real numbers. $$ \sqrt[5]{\frac{2}{243}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{\sqrt[5]{2}}{3} \).

Step 1: Identify the given expression

The expression provided is \( \sqrt[5]{\frac{2}{243}} \). This is a fifth root of a fraction.

Step 2: Break down the fraction under the root

Recognize that the expression can be written as \( (2/243)^{1/5} \). Break down the denominator: \( 243 = 3^5 \). So, the expression becomes \( \sqrt[5]{\frac{2}{3^5}} \).

Step 3: Apply the root to numerator and denominator

Using the property \( \sqrt[b]{\frac{a}{b}} = \frac{\sqrt[b]{a}}{\sqrt[b]{c}} \), rewrite the expression as \( \frac{\sqrt[5]{2}}{\sqrt[5]{3^5}} \).

Step 4: Simplify the fifth root of the denominator

Since \( \sqrt[5]{3^5} = 3 \), simplify the expression to \( \frac{\sqrt[5]{2}}{3} \).

Step 5: Expression in simplest form

The expression \( \frac{\sqrt[5]{2}}{3} \) is already in its simplest form because \( \sqrt[5]{2} \) cannot be simplified further.

Key Concepts

Addition of Radicals

Adding radicals might seem confusing at first, but it resembles adding common numbers or like terms. The key rule is that you can only add radicals directly if they have the same index and radicand. This way, they are considered 'like' radicals, similar to how you might combine like terms in algebra.

Here's a simple approach to add radicals:

• Look for radicals with the same index and radicand (e.g., \( \sqrt{3} + 2\sqrt{3} \)).
• Add or subtract the coefficients, keeping the radical part unchanged (so, \( 1\sqrt{3} + 2\sqrt{3} = 3\sqrt{3} \)).

If the radicals are not alike, you cannot combine them directly, and you must simplify each expression separately.

For example, consider \( \sqrt[3]{5} + \sqrt[3]{7} \). Since the radicands (5 and 7) are different, you can't add them directly. Instead, if possible, simplify each radical individually and see if any become alike.

Subtraction of Radicals

Subtraction of radicals follows a similar process as addition. You can only subtract radicals that are 'like', meaning they share the same index and radicand.

Here is a step-by-step guide on how to subtract radicals:

• Identify radicals that have the same index and radicand (e.g., \( 5\sqrt{2} - 3\sqrt{2} \)).
• Subtract the coefficients (so, \( 5\sqrt{2} - 3\sqrt{2} = 2\sqrt{2} \)).

Remember to always check if the radicals can be simplified before performing subtraction. For instance, if you have radicals like \( \sqrt{50} \) and \( \sqrt{18} \), you might need to simplify them first to \( 5\sqrt{2} \) and \( 3\sqrt{2} \) before realizing they can be combined.

When dealing with different radicals, as in \( \sqrt[4]{16} - \sqrt{9} \), simplify each part if possible, then see if subtraction can be carried out. Here, \( \sqrt[4]{16} = \sqrt{2} \) and \( \sqrt{9} = 3 \), so they cannot be directly subtracted.

Rationalizing the Denominator

Rationalizing the denominator is a fundamental process in algebra, meant to eliminate radicals in the denominator of a fraction.

Here's how you can approach rationalizing:

• If the denominator is a simple square root, multiply both the numerator and the denominator by that root. For example, to rationalize \( \frac{1}{\sqrt{3}} \), multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \) to get \( \frac{\sqrt{3}}{3} \).
• If you have an expression with a radical plus or minus another term, like \( 1 + \sqrt{3} \), multiply by the conjugate, which in this case is \( 1 - \sqrt{3} \).
• The product will eliminate the radical after applying the difference of squares for products like \( (1 + \sqrt{3})(1 - \sqrt{3}) = 1 - 3 \).

In more complex radicals like cube roots or higher, carefully apply similar strategies, perhaps by identifying what to multiply to achieve a complete integer power. Rationalizing helps in obtaining a more manageable form, free from radicals in the denominator.