Problem 58 - Algebra A Combined Function

Question

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt[3]{\frac{25}{2}}\)

Step-by-Step Solution

Answer

Rationalized result: \( \frac{5}{\sqrt[3]{10}} \).

Step 1: Identify the Problem

The problem asks us to rationalize the numerator of the expression \( \sqrt[3]{\frac{25}{2}} \). The numerator here is 25, under a cube root.

Step 2: Convert to Indexed Expression

Express \( \sqrt[3]{\frac{25}{2}} \) using exponents instead of roots for easier manipulation. This becomes \( \left( \frac{25}{2} \right)^{\frac{1}{3}} \).

Step 3: Multiply to Clear Cube Root from Numerator

To rationalize, we need 25 to become a perfect cube. Multiply both numerator and denominator by \( \sqrt[3]{5} \) such that \( 25 \cdot 5 = 125 \), a perfect cube.

Step 4: Perform Multiplication

Multiply the fraction by \( \frac{\sqrt[3]{5}}{\sqrt[3]{5}} \):\[\frac{\sqrt[3]{25} \cdot \sqrt[3]{5}}{\sqrt[3]{2} \cdot \sqrt[3]{5}}\]. This yields:\[\frac{\sqrt[3]{125}}{\sqrt[3]{10}}\].

Step 5: Simplify Perfect Cube

Since \(125 = 5^3\), \( \sqrt[3]{125} = 5 \). Therefore, the expression simplifies to:\[\frac{5}{\sqrt[3]{10}}\].

Key Concepts

Cube Roots

Cube roots are a type of radical expression where you're looking for a number that, when used three times in a multiplication, gives you the original number. For instance, the cube root of 8 is 2, because when you multiply 2 by itself three times (2 × 2 × 2), you get 8. This is expressed as \( \sqrt[3]{8} = 2 \). The cube root symbol \( \sqrt[3]{\cdot} \) is used, and finding a cube root can also be represented using exponents, such as \( x^{\frac{1}{3}} \). Finding cube roots can be particularly useful when simplifying expressions with fractional bases, as demonstrated in the original exercise. Here, the goal of rationalizing the numerator is to get a whole number, making further calculations easier.

Indexed Expressions

Indexed expressions utilize the power of exponents to represent radicals in an alternative form. The base, raised to an exponent, describes how many times the base is used in a multiplication operation. In our case, using \( \sqrt[3]{\frac{25}{2}} \) as \( \left( \frac{25}{2} \right)^{\frac{1}{3}} \) helps in operations like multiplication and division.When dealing with these expressions, they offer substantial flexibility:

• They allow converting complex root problems into more manageable, algebraic forms.
• Expressions are easy to manipulate, since basic rules of exponents apply, like \( x^a \times x^b = x^{a+b} \).

Understanding indexed expressions aids in recognizing patterns in problems and applying consistent methods of solving, such as rationalizing numerators by dealing with the roots as exponents.

Perfect Cubes

A perfect cube is a number that can be expressed as the cube of an integer, which means it's the result of multiplying an integer by itself twice more. For example, 27 is a perfect cube because it's \( 3^3 \) (or 3 multiplied by itself three times). Recognizing perfect cubes helps in simplifying cube roots, as they result in whole numbers.In rationalizing numerators, creating perfect cubes in the numerator allows for cancellation with the cube root, simplifying the expression. In our exercise, multiplying 25 by 5 to get 125, which is \( 5^3 \), allows the cube root of the numerator to be simplified to 5. This step systematically brings us closer to obtaining a simpler rational expression.

Multiplication of Fractions

Multiplying fractions involves a straightforward process of multiplying the numerators together and the denominators together. This method aligns perfectly with indexed expressions and rationalization practices. In more complex expressions involving cube roots, this practice can simplify seemingly challenging expressions.In the exercise, to rationalize \( \sqrt[3]{\frac{25}{2}} \), we used the identity \( \frac{\sqrt[3]{5}}{\sqrt[3]{5}} \). Multiply both numerator and denominator by \( \sqrt[3]{5} \), which led us to convert the numerator into a perfect cube.

• This approach maintains the equivalency of the fraction, as multiplying by 1 does not change the value.
• The multiplication helps in eliminating the radical from the numerator, paving the path for simplification.

Through multiplication, fractions become simpler to handle, especially when tied with radicals, making calculations and further algebraic manipulation more manageable.