Problem 31 - Algebra 2

Question

Rationalize the denominator of each expression. Assume that all variables are positive. $$ \frac{\sqrt[4]{2}}{\sqrt[4]{5}} $$

Step-by-Step Solution

Answer

The rationalized form of the given expression is \( \frac{\sqrt[4]{250}}{5} \)

Step 1: Identify the denominator and its root

The denominator of the given expression is \( \sqrt[4]{5} \). This is a fourth root.

Step 2: Determine the rationalizing factor

To remove the fourth root from the denominator, we will need to multiply it by the same root raised to the power of 3. So, the rationalizing factor is \( (\sqrt[4]{5})^3 \), which is simply \( \sqrt[4]{125} \).

Step 3: Multiply numerator and denominator by rationalizing factor

Multiply both, the numerator and denominator of the original expression, by \( \sqrt[4]{125} \).\n So, \[ \frac{\sqrt[4]{2}}{\sqrt[4]{5}} = \frac{\sqrt[4]{2} * \sqrt[4]{125}}{\sqrt[4]{5} * \sqrt[4]{125}} \]

Step 4: Simplify the expression

Simplify the expression obtained from step 3 to get the final answer.\n So, \[ \frac{\sqrt[4]{250}}{\sqrt[4]{625}} \] Simplifying further, we get \[ \frac{\sqrt[4]{250}}{5} \]

Key Concepts

Fourth Roots

A fourth root is a special type of root where you are looking for a number that, when multiplied by itself four times, gives you the original number. It's similar to a square root but to the power of four. For example, the fourth root of 16 is 2, because

• \(2 \times 2 \times 2 \times 2 = 16\)\.
• Mathematically, this can be represented as \( \sqrt[4]{16} = 2\).

When working with fourth roots in algebraic expressions, particularly in denominators, simplifying is often necessary. This involves rationalizing the denominator. You multiply the numerator and denominator by a factor that will eliminate the root in the denominator, making the expression easier to work with and understand.

Simplifying Expressions

Simplifying expressions involves reducing them to their most concise and simplest form. This process often includes eliminating complex or unnecessary elements of the expression, such as roots or fractions that can obscure the value you are calculating.

• In the context of the problem, simplifying means getting rid of the fourth root on the denominator.
• To do this, multiply the entire expression by a factor that will turn the complex denominator into a rational number.
• After rationalizing the denominator, simplify the resulting expression by performing the multiplication and reducing where possible.

Ultimately, this process helps simplify calculations and allows for easier interpretation of the results.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. Variables are symbols that stand in for unknown or arbitrary numbers, and they are typically represented by letters like \(x\), \(y\), or in this case, roots such as \(\sqrt[4]{5}\).

• Algebraic expressions can describe a wide range of mathematical phenomena and are essential in forming equations and inequalities.
• In exercises involving rationalizing denominators, algebraic expressions often include radical expressions or roots. These require special techniques to simplify.
• The goal when working with algebraic expressions is often to simplify them, either by combining like terms, factoring, or rationalizing as discussed in this context.

These tasks are central to algebra and help students develop problem-solving skills they will use in more advanced mathematics.