Problem 85 - Algebra A Combined Function

Question

Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the radical expression. \(\frac{9}{\sqrt[3]{5}}\)

Step-by-Step Solution

Answer

Multiply both by \((\sqrt[3]{5})^2\).

Step 1: Understand the Problem

We want to eliminate the cube root from the denominator \(\sqrt[3]{5}\) of the fraction \(\frac{9}{\sqrt[3]{5}}\) by converting it into a rational number.

Step 2: Identify the Multiplier

To rationalize the denominator \( \sqrt[3]{5} \), you need to multiply by something that makes the result a whole number. If you multiply \(\sqrt[3]{5}\) by itself twice more (to make it \((\sqrt[3]{5})^3 = 5\)), it will become rational. So we multiply both the numerator and denominator by \((\sqrt[3]{5})^2\).

Step 3: Perform the Multiplication

Multiply both the numerator and the denominator by \((\sqrt[3]{5})^2\): \[\frac{9}{\sqrt[3]{5}} \times \frac{(\sqrt[3]{5})^2}{(\sqrt[3]{5})^2} = \frac{9 \times (\sqrt[3]{5})^2}{5}\].

Step 4: Simplify the Expression

After multiplying, the expression simplifies to \[\frac{9 (\sqrt[3]{5})^2}{5}\]. The denominator is now rational.

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that include roots, such as square roots, cube roots, or even higher order roots. These expressions use the radical symbol (√) or fractional exponents to indicate operations involving roots. Understanding how to manipulate and simplify radical expressions is a fundamental skill in algebra.
When working with radical expressions, it's important to know how to simplify them and perform operations like addition, subtraction, multiplication, and division.

• Simplifying a radical expression involves rewriting it with the smallest possible radicand (the number inside the radical sign) by factoring out perfect squares, cubes, or higher powers.
• Operations with radical expressions often require converting them into equivalent forms with rational numbers when possible, particularly in denominators.
• Rationalizing the denominator is a common technique used to eliminate radicals from the denominator of a fraction, often by multiplying by an appropriate form of 1, such as the conjugate or a higher power of the radicand, effectively removing the radical.

Being comfortable with these concepts simplifies further mathematical calculations and enhances problem-solving skills.

Cube Roots

Cube roots are a specific type of radical expression that involve taking a number to the third root. The cube root of a number is a value which, when multiplied by itself twice, gives the original number.
It's important to distinguish cube roots from square roots, as the operations and results differ significantly.

• To find the cube root of a number, you are essentially solving for the value of a number written as \( \sqrt[3]{x} \), where #x# is the number you're taking the cube root of.
• Cube roots of perfect cubes are whole numbers. For instance, the cube root of 8 is 2, since \( 2 \times 2 \times 2 = 8 \).
• In expressions like \( \frac{9}{\sqrt[3]{5}} \), the goal often involves eliminating the cube root from the denominator, which requires rationalizing the expression—converting it to a fraction with a whole number in the denominator.

Mastering operations with cube roots is not only helpful for algebra, but it's also crucial for understanding more complex equations in higher mathematics.

Multiplying Fractions

Multiplying fractions is a fundamental arithmetic operation that is simpler than it often appears. To multiply fractions, you need to follow a straightforward set of steps:

• Multiply the numerators (the numbers on top) together to find the numerator of the answer.
• Multiply the denominators (the numbers on the bottom) together to find the denominator of the answer.
• Simplify the resulting fraction by canceling any common factors in the numerator and denominator, making sure to reduce it to its simplest form.

Understanding this process is essential when working with expressions that include fractions, such as \( \frac{9}{\sqrt[3]{5}} \times \frac{(\sqrt[3]{5})^2}{(\sqrt[3]{5})^2} \). In such scenarios, multiplying by the appropriate radical expression helps in rationalizing the denominator, ultimately transforming it into a fraction with a rational whole number in the denominator.
Once you are comfortable with multiplying fractions, ensuring they're reduced to their simplest forms becomes second nature, allowing for more fluid calculations. Filled with practical use, this skill is valuable not only in algebra but in everyday numeracy and problem-solving tasks.