Problem 67 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rationalize the denominator of the expression. $$ \frac{y}{\sqrt[3]{x^{2} z}} $$

Step-by-Step Solution

Answer

The rationalized expression is: \(\frac{yz^{2}}{z\sqrt[3]{x^{2}}}\)

Step 1: Identifying the powers of the variables under the radical

First, let's identify the powers of the variables, x and z, under the cube root. In our expression, the powers under the cube root are x² and z.

Step 2: Finding the least common multiple (LCM) of the powers

In the cube root (denominator), we have x² and z. To eliminate the cube root, we need to find the \(LCM \ of \ the \ powers\), given by \(3\). So, we need to multiply the denominator by \(x^{(3-2)}z^{(3-1)}\), which simplifies to \(xz^{2}\).

Step 3: Rationalizing the denominator

Since we want to eliminate the cube root in the denominator fully, we need to multiply both the numerator and denominator of our expression by \(xz^{2}\). By doing this, we will have: $$ \frac{y}{\sqrt[3]{x^{2}z}} \times \frac{xz^{2}}{xz^{2}} = \frac{yxz^{2}}{\sqrt[3]{x^{5}z^{3}}} $$

Step 4: Simplifying the denominator

Now, we can simplify the denominator as we have rationalized it: $$ \frac{yxz^{2}}{\sqrt[3]{x^{5}z^{3}}} = \frac{yxz^{2}}{xz\sqrt[3]{x^{2}}} $$

Step 5: Final simplified expression

Finally, we can simplify the expression by canceling out the common terms in the numerator and denominator: $$ \frac{yxz^{2}}{xz\sqrt[3]{x^{2}}} = \frac{y\cancel{x}z^{2}}{\cancel{x}z\sqrt[3]{x^{2}}} = \frac{yz^{2}}{z\sqrt[3]{x^{2}}} $$ So, the simplified and rationalized expression is: $$ \frac{yz^{2}}{z\sqrt[3]{x^{2}}} $$

Key Concepts

Cube Root Simplification

The process of cube root simplification is key when dealing with radical expressions involving variables. A cube root, written as \( \sqrt[3]{n} \), represents a value that, when multiplied by itself three times, gives the number \(n\). Simplifying a cube root involves finding an integer or polynomial that balances the inside expression to a perfect cube. For example, \( \sqrt[3]{x^3} \) simplifies to \(x\), but \( \sqrt[3]{x^2} \) requires multiplication by \( \sqrt[3]{x} \) to reach \(x\).

When simplifying cube roots in fractions, the goal is often to 'rationalize the denominator,' which means to eliminate the radical from the denominator. If the denominator is not a perfect cube, we must multiply the fraction by a form of one that will create a perfect cube under the radical to simplify effectively.

Least Common Multiple in Algebra

In algebra, the least common multiple (LCM) is used to find the smallest common multiple of algebraic expressions, such as variables with exponents. For example, \(LCM(x^2, z)\) means finding the smallest power of \(x\) and \(z\) that each term can divide into without leaving a remainder. The LCM plays a crucial role in simplifying expressions, particularly when adding, subtracting, or comparing algebraic fractions with different denominators. In our exercise, we used the LCM concept to rationalize the denominator by finding the appropriate exponents of \(x\) and \(z\) that when multiplied with the existing terms, would create a perfect cube – thus allowing us to simplify by taking the cube root.

Radical Expressions

• Radical expressions involve roots, such as square roots, cube roots, and higher-order roots.
• One frequently encountered operation is to combine or simplify these expressions, which often include adding or subtracting like terms or multiplying and dividing by conjugates to rationalize denominators.
• When dealing with cube roots, it is important to remember that you can only combine like terms, which are terms with the same index and radicand, or the number under the radical.

Radical expressions must sometimes be manipulated to perform operations like addition or rationalization, requiring a good understanding of radical properties and how to apply them to algebraic terms.

Simplifying Algebraic Fractions

Algebraic fractions, similar to numerical fractions, consist of a numerator and a denominator comprising algebraic expressions. The simplification of these fractions often includes factoring, finding common denominators, and canceling like terms. Simplifying an algebraic fraction can make it much easier to work with, especially when solving equations or inequalities. In our example, once we rationalized the denominator and created a perfect cube, we simplified the resulting algebraic fraction by canceling out terms that appeared in both the numerator and the denominator. This cancellation is possible because of the fundamental property that \(\frac{a}{b}\times\frac{b}{a} = 1\), which is applied to terms in algebraic fractions, thereby reducing them to their simplest form.