Problem 87 - Elementary and Intermediate Algebra

Question

Rationalize each denominator. All variables represent positive real numbers. $$ \frac{\sqrt[4]{2}}{\sqrt[4]{3 t^{2}}} $$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{(54 t^6)^{1/4}}{3t^2} \)."

Step 1: Recognize the Problem

We have a fourth root in the denominator, \( \frac{\sqrt[4]{2}}{\sqrt[4]{3t^2}} \). The goal is to eliminate the radical in the denominator by rationalizing it.

Step 2: Identify the Conjugate Factor

To rationalize \( \sqrt[4]{3t^2} \), we need its fourth power. We will multiply both the numerator and the denominator by \( \sqrt[4]{3t^2}^3 = (3t^2)^{3/4} \) to make the denominator a power of \( (3t^2) \).

Step 3: Multiply Numerator and Denominator

Multiply both the numerator and the denominator by \( (3t^2)^{3/4} \):\[ \frac{\sqrt[4]{2}}{\sqrt[4]{3t^2}} \times \frac{(3t^2)^{3/4}}{(3t^2)^{3/4}} = \frac{\sqrt[4]{2} \times (3t^2)^{3/4}}{\sqrt[4]{3t^2} \times (3t^2)^{3/4}} \]

Step 4: Simplify the Denominator

The denominator becomes \((3t^2)^{4/4} = 3t^2\), since we're raising it to a full power of 1 by adding exponents.

Step 5: Simplify the Numerator

The numerator becomes \((2)^{1/4} \times (3t^2)^{3/4} = (2 \times 3^3 t^6)^{1/4} = (54 t^6)^{1/4}\).

Step 6: Final Answer

The rationalized form of the expression is \( \frac{(54 t^6)^{1/4}}{3t^2} \). This removes the radical from the denominator and rationalizes the expression.

Key Concepts

Algebraic Expressions

In mathematics, algebraic expressions form the foundation upon which we build more complex equations and functions. An algebraic expression consists of numbers, variables, and operators. Variables are symbols, often letters, that stand in for unknown values and can change depending on the equation or context. For example, in the algebraic expression \( 3x + 7 \), 3 and 7 are numbers, \( x \) is a variable, and the plus sign is the operator. Such expressions can be manipulated using rules of arithmetic to solve for variables or to simplify them. Understanding algebraic expressions helps us perform operations like addition, subtraction, multiplication, and division involving unknown quantities. In the exercise provided, algebraic expressions appear in both the numerator and the denominator, where we see \( \sqrt[4]{2} \) and \( \sqrt[4]{3t^2} \). Our task is to manipulate these expressions to achieve a simpler form, which often involves rationalizing the denominator.

Fourth Root

The concept of the fourth root is essential when dealing with radicals in algebra. The fourth root of a number \( x \) is any number that, when raised to the power of four, equals \( x \). We express this as \( \sqrt[4]{x} \). It is similar to the more familiar square root, but here we are working with powers of four.For instance, for the number 16, its fourth root is 2 because \( 2^4 = 16 \). Likewise, the fourth root of an algebraic expression \( 3t^2 \) is a value which when raised to the fourth power gives \( 3t^2 \).Understanding this concept is key to simplifying expressions where fourth roots are present, especially in efforts to eliminate them from the denominator. When you remove a fourth root from the denominator, you must multiply by enough of the radical to make the power add up to a whole number, which is exactly what the exercise requires.

Simplifying Radicals

Simplifying radicals is an important skill in algebra that allows us to express complex radical expressions in their simplest form. This involves manipulating the expression so that it no longer contains any roots in the denominator, which makes the expression more conventional and easier to interpret.When simplifying the expression \( \frac{\sqrt[4]{2}}{\sqrt[4]{3t^2}} \), we focus on the denominator \( \sqrt[4]{3t^2} \). By multiplying both the numerator and the denominator by \( (3t^2)^{3/4} \), we convert \( \sqrt[4]{3t^2} \) to a whole power, resulting in \( 3t^2 \). This rationalizes the denominator.In the numerator, multiplying \( \sqrt[4]{2} \) by \( (3t^2)^{3/4} \) transforms it into \( (54t^6)^{1/4} \), which we simplify as much as possible, leading to a cleaner, more manageable expression. Simplifying radicals involves understanding how exponents and roots work together, allowing us to transform and rationalize expressions effectively.