Problem 88 - Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt[4]{3}}{\sqrt[4]{5 b^{3}}} $$

Step-by-Step Solution

Answer

Rationalized expression: \(\frac{\sqrt[4]{15b}}{5b}\).

Step 1: Understanding the Problem

The problem requires us to rationalize the denominator of the expression \(\frac{\sqrt[4]{3}}{\sqrt[4]{5b^{3}}}\). Rationalizing the denominator means rewriting the expression so that the denominator is a rational number or has no roots.

Step 2: Identifying the Denominator Problem

The denominator \(\sqrt[4]{5b^3}\) is a fourth root, which is an irrational expression. This needs to be rationalized.

Step 3: Constructing the Rationalizing Factor

To rationalize \(\sqrt[4]{5b^3}\), we look for a number we can multiply both the numerator and the denominator by that will result in an integer when raised to the fourth power. In this case, multiply by \(\sqrt[4]{5b^3}\) such that it becomes \(5b^3\) in the fourth power. So, multiply by \(\sqrt[4]{5b}\).

Step 4: Multiply Numerator and Denominator by the Rationalizing Factor

Multiply both the numerator and the denominator by \(\sqrt[4]{5b}\) : \[\frac{\sqrt[4]{3}}{\sqrt[4]{5b^3}} \cdot \frac{\sqrt[4]{5b}}{\sqrt[4]{5b}}\] This gives:\[\frac{\sqrt[4]{15b}}{\sqrt[4]{5^2 b^4}} = \frac{\sqrt[4]{15b}}{5b}\]

Step 5: Simplify the Expression

Now, the denominator \(5b\) is a rational number, and the expression \(\frac{\sqrt[4]{15b}}{5b}\) is fully simplified.

Key Concepts

Fourth roots

Fourth roots are similar to square roots but instead, they involve finding a number that, when raised to the power of four, gives the original number.
For example, in the expression \( \sqrt[4]{3} \), we are looking for a number such that when multiplied by itself four times ( i.e., \( x^4 \)), the result would be 3.
This concept is not just limited to numbers; variables can be under fourth roots as well, as seen in \( \sqrt[4]{5b^3} \).

• Understanding fourth roots: the output is a number or expression that raised to the power of four equals the number inside the root.
• They are important in various algebraic manipulations, especially when simplifying or rationalizing expressions.

Being comfortable with fourth roots is essential for tasks such as rationalizing denominators in algebra and dealing with complex numbers in higher mathematics.

Irrational expressions

Irrational expressions involve numbers or expressions that cannot be expressed as simple fractions.
They often include roots of numbers which result in endless non-repeating decimals. Examples include \( \sqrt{2} \) or \( \sqrt[4]{5} \).
In the problem, the denominator \( \sqrt[4]{5b^3} \) represents an irrational expression that we wish to change into a more manageable rational form.

• A rational number either has a finite decimal expansion or repeats a sequence (e.g., 0.333...).
• An irrational expression can often be transformed by multiplying by a carefully chosen factor to "rationalize" it.

By altering irrational expressions into rational ones, we can simplify expressions and make them easier to work with. This is particularly useful in solving equations or integrating radical expressions into broader mathematics problems.

Radical expressions

Radical expressions consist of any expression that contains a root, such as a square root, cube root, or fourth root.
The general form is \( \sqrt{a} \) or for other roots \( \sqrt[n]{a} \). Radical expressions like \( \sqrt[4]{3} \) and \( \sqrt[4]{5b^3} \) might initially seem challenging, but they can often be simplified or transformed into a more convenient form.

• Cleaning up radicals: this involves manipulating the expression so that the root is no longer in the denominator — a process known as rationalizing.
• Multiplying by appropriate radical expressions can simplify the irrational roots.
• Key goal in simplification: to represent complex forms in a way that's easy to understand and manipulate further.

Mastering the manipulation of radical expressions is crucial for solving equations, graphing functions, and exploring advanced mathematical concepts in calculus and beyond.