Problem 29 - Intermediate Algebra

Question

Rationalize each denominator. See Examples 1 through 3. $$ \sqrt[4]{\frac{81}{8}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{3 \sqrt[4]{2}}{2} \).

Step 1: Simplify the Radical Expression

First, let's simplify the given radical expression \( \sqrt[4]{\frac{81}{8}} \). This can be rewritten as \( \frac{\sqrt[4]{81}}{\sqrt[4]{8}} \). Simplifying the numerator, we have \( \sqrt[4]{81} = 3 \) since \( 3^4 = 81 \).

Step 2: Rationalize the Denominator

We need to eliminate the fourth root from the denominator. To do this, we multiply both the numerator and the denominator by \( \sqrt[4]{8^3} \), because \( (\sqrt[4]{8})^4 = 8 \). Thus, we multiply by \( \sqrt[4]{512} \).

Step 3: Multiply and Simplify

Multiply the expression by \( \frac{\sqrt[4]{512}}{\sqrt[4]{512}} \): \[ \frac{3 \cdot \sqrt[4]{512}}{\sqrt[4]{8} \cdot \sqrt[4]{512}} = \frac{3 \sqrt[4]{512}}{8} \]. The denominator simplifies to 8 since \( \sqrt[4]{8 \cdot 512} = \sqrt[4]{4096} = 8 \).

Step 4: Final Simplification

Now, check if the expression can be simplified further. The numerator is \( 3 \sqrt[4]{512} \). Since 512 can be further decomposed to \( 2^9 \), we have: \( \sqrt[4]{512} = 2^{9/4} = 2^2 \cdot 2^{1/4} = 4 \cdot \sqrt[4]{2} \). So, the expression becomes \( \frac{3 \cdot 4 \cdot \sqrt[4]{2}}{8} = \frac{12 \sqrt[4]{2}}{8} = \frac{3 \sqrt[4]{2}}{2} \) after simplifying.

Key Concepts

Simplifying Radical Expressions

Simplifying radical expressions is a crucial skill in mathematics, especially when dealing with radicals and roots. A radical expression includes a root, such as a square root or fourth root. The process of simplifying involves reducing the expression to its simplest form, much like reducing fractions.
To begin with, break down the components of the radical. For example, when simplifying \( \sqrt[4]{81} \), it's valuable to recognize that 81 is a perfect fourth power. Since \( 3^4 = 81 \), we know \( \sqrt[4]{81} = 3 \). By identifying perfect powers, the simplification becomes straightforward. Similarly, when handling the denominator \( \sqrt[4]{8} \), it helps to express 8 in terms of its prime factors, \( 2^3 \). Recognizing these powers helps in applying further steps.
Always aim to express the radical in its simplest power terms, and try to identify any numerical relationships that allow for the extraction of whole numbers.

Fourth Roots

Fourth roots are less common than square or cube roots, but they function similarly. Taking the fourth root of a number involves finding a value that, when raised to the fourth power, equals the original number. For example, \( \sqrt[4]{81} \) asks which number raised to the fourth power equals 81. Since \( 3^4 = 81 \), the fourth root of 81 is 3.
Understanding fourth roots requires familiarity with exponents. You can think of fourth roots as the inverse operation of raising a number to the fourth power. If \( x^4 = y \), then \( \sqrt[4]{y} = x \). This relationship is integral in both simplifying expressions and solving equations involving fourth roots.
Remember that fourth roots can be broken down for more complex numbers by expressing numbers in terms of their prime factors, which can help identify components that can be extracted outside the root as integers.

Fraction Simplification

Fraction simplification is often needed after performing operations on fractions, such as multiplying both the numerator and denominator by a common factor. The objective is to make the fraction as simple as possible while retaining its value.
When rationalizing denominators, it's essential to perform equivalent operations on both the numerator and the denominator to eliminate any roots in the denominator. For instance, multiplying both parts by \( \sqrt[4]{512} \) simplifies the denominator of \( \sqrt[4]{8} \), which is crucial for making the expression easier to handle.
In the example provided, all resulting expressions are simplified by ensuring that any common factors between the numerator and the denominator are canceled out. This is how the expression \( \frac{12 \sqrt[4]{2}}{8} \) was simplified to \( \frac{3 \sqrt[4]{2}}{2} \). Look for common factors that can be divided out.

Radical Operations

Performing operations with radicals might seem complex initially, but understanding the properties of radicals can make the process manageable. Operations involve various actions like multiplying, dividing, and rationalizing expressions with radicals.
In rationalizing denominators, the main goal is to remove the radical so that the denominator is a more manageable number. This typically involves multiplying by a conjugate or some expression that will eliminate the root. Multiplying both the numerator and the denominator by \( \sqrt[4]{512} \) is an example of such a strategy.
A key aspect of operations with radicals is maintaining equality by performing the same action to both the numerator and the denominator. This keeps the expression equivalent to its original form while making it easier to work with. Always remember to simplify the resulting expressions to their basic form, easing further mathematical operations. Recognizing how to manipulate and simplify radical components is crucial for mastering radical operations.