Problem 59 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rationalize the denominator of the expression. $$ \frac{1}{\sqrt[3]{x}} $$

Step-by-Step Solution

Answer

The short answer based on the given step-by-step solution is: To rationalize the denominator of the expression \( \frac{1}{\sqrt[3]{x}} \), multiply both the numerator and denominator by \(\sqrt[3]{x^{2}}\), and simplify the resulting expression to obtain \( \frac{\sqrt[3]{x^2}}{x} \).

Step 1: Identify the given expression.

The expression given to us is \( \frac{1}{\sqrt[3]{x}} \).

Step 2: Determine what to multiply with numerator and denominator to get rid of the cube root.

We need to find a way to get rid of the cube root in the denominator. Since it is a cube root, we can multiply the expression by the square of the denominator, which is \(\sqrt[3]{x^{2}}\).

Step 3: Multiply the numerator and denominator by \(\sqrt[3]{x^{2}}\).

Multiplying both numerator and denominator by \(\sqrt[3]{x^{2}}\), we get: \[ \frac{1}{\sqrt[3]{x}} \times \frac{\sqrt[3]{x^{2}}}{\sqrt[3]{x^{2}}} = \frac{\sqrt[3]{x^{2}}}{\sqrt[3]{x^3}}. \]

Step 4: Simplify the expression.

Now, we need to simplify the expression. In the denominator, we have a cube root of a cube, which cancels out: \[ \frac{\sqrt[3]{x^{2}}}{\sqrt[3]{x^3}} = \frac{\sqrt[3]{x^2}}{x}. \] So, the rationalized expression is \( \frac{\sqrt[3]{x^2}}{x} \).

Key Concepts

Cube Root

The cube root is a way to find a number that, when multiplied by itself three times, gives you the original number. It's the opposite of cubing a number. For example, the cube root of 27 is 3, because when you multiply 3 by itself three times (3 × 3 × 3), you get 27. In mathematical notation, the cube root of a number is written as \( \sqrt[3]{x} \), where \( x \) is the number we are finding the cube root of.

When dealing with algebraic expressions that include cube roots, especially when they appear in the denominator, it's important to eliminate them so the expression becomes easier to work with. This process is known as "rationalizing the denominator." For cube roots, this often involves multiplying by a suitable expression that results in a perfect cube under the cube root, simplifying our calculations.

Expression Simplification

Simplifying an expression involves combining and reducing it to its simplest form. This means getting rid of any complex roots, common factors, or redundant terms.

To simplify the expression \( \frac{1}{\sqrt[3]{x}} \), you should ensure you eliminate the cube root in the denominator. You achieve this by finding an expression that, when multiplied with the denominator, yields a perfect cube. In our case, multiplying by \( \sqrt[3]{x^2} \) effectively turns \( \sqrt[3]{x^3} \) into \( x \).

When you eliminate the cube root from the denominator, the resulting fraction is \( \frac{\sqrt[3]{x^2}}{x} \). This expression is in its simplest form because it doesn't have roots in its denominator, making it neater and more versatile for mathematical operations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules and properties. The key steps include multiplying, factoring, expanding, or reducing expressions to simplify them.

In rationalizing a denominator that contains a cube root, algebraic manipulation is crucial. You first multiply the numerator and the denominator by the same expression to create a perfect cube in the denominator. Here, multiplying by \( \sqrt[3]{x^2} \) served that purpose. As a result, you achieve a simplified form that is much easier to work with.

The skills used in these manipulations include understanding exponents, roots, and basic fractional arithmetic. This approach not only helps in rationalizing denominators but also in various other algebraic tasks like simplifying expressions or solving equations. Mastering these manipulative techniques can greatly enhance problem-solving abilities in mathematics.