Problem 25 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{\sqrt[3]{2 y^{2}}}{\sqrt[3]{9 x^{2}}}\)

Step-by-Step Solution

Answer

The rationalized expression is \(\frac{\sqrt[3]{162y^2x^4}}{9x^2}\).

Step 1: Identify the Expression

We are given the expression \(\frac{\sqrt[3]{2 y^{2}}}{\sqrt[3]{9 x^{2}}}\). The goal is to rationalize its denominator. A rational denominator contains no radicals.

Step 2: Determine the Rationalization Factor

To rationalize \(\sqrt[3]{9x^2}\), we need a factor that makes the cube root a whole number. This factor would be \(\sqrt[3]{9x^2}\) such that when multiplied, the result is \(9x^2\). Therefore, we multiply by \(\sqrt[3]{(9x^2)^2}\), which is \(\sqrt[3]{81x^4}\).

Step 3: Multiply by the Rationalizing Factor

We multiply the numerator and the denominator by \(\sqrt[3]{81x^4}\):\[\frac{\sqrt[3]{2y^2} \times \sqrt[3]{81x^4}}{\sqrt[3]{9x^2} \times \sqrt[3]{81x^4}}\]

Step 4: Simplify the Denominator

In the denominator, \(\sqrt[3]{9x^2} \times \sqrt[3]{81x^4} = \sqrt[3]{(9x^2)(81x^4)} = \sqrt[3]{729x^6} = 9x^2\). The denominator is now rational and simplified.

Step 5: Simplify the Numerator

In the numerator, use the property of radicals: \(\sqrt[3]{2y^2} \times \sqrt[3]{81x^4} = \sqrt[3]{(2y^2)(81x^4)} = \sqrt[3]{162y^2x^4}\).

Step 6: Write the Final Expression

The rationalized expression becomes:\[\frac{\sqrt[3]{162y^2x^4}}{9x^2}\]This is the expression with the denominator rationalized.

Key Concepts

Cube Roots

A cube root, denoted by \( oot{3}{x}\), is a number that, when multiplied by itself three times, gives the original number \((x)\). In other words, the cube root undoes cubing a number. Let's take the example \( oot{3}{8} = 2\). This is because 2 multiplied by itself twice more (i.e., 2 \(\times\) 2 \(\times\) 2) equals 8.
Cube roots can be extended beyond whole numbers. For instance, with cube roots involving variables like \(x\), \( oot{3}{x^3} = x\) because \(x \times x \times x = x^3\).
Understanding cube roots is essential when working with radical expressions, as it helps simplify expressions and manage them, especially when dealing with algebraic expressions involving powers of variables.

Simplifying Radicals

Simplifying radicals refers to the process of making a radical expression more manageable by eliminating radicals in denominators. This often involves reducing the number of terms under the radical. For example, the radical \( oot{3}{8x^3}\) can be simplified by factoring out perfect cubes.
The general approach for simplifying involves dividing the radicand (the number under the root) into its factor components. This breaks down the expression to its simplest form, while ensuring the variables are positive and the radical expression is clear.
In simplifying cube roots, it’s crucial to look for terms that can be expressed as cubes. This makes the process easier and helps in rationalizing the denominator, which is often the ultimate goal in these expressions. Simplifying radicals forms the cornerstone of understanding more complex expressions involving roots.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and operations (addition, subtraction, multiplication, and division). These expressions form the heart of algebra as they represent quantities without exact values, offering us solutions to equations or inequalities.
When dealing with radicals within algebraic expressions, like \(rac{\sqrt[3]{2y^2}}{\sqrt[3]{9x^2}}\), it’s important to understand how to manipulate them. This could involve rationalizing denominators or simplifying radicals, both of which are valuable algebraic skills.
Algebraic expressions can become complex, but breaking them down into smaller parts or factors helps. Recognizing patterns through structures existing within these expressions, such as cubes or squares, enables easier manipulation and simplification. It provides a pathway to solving equations or simplifying expressions involving radicals and their roots.

Properties of Exponents

The properties of exponents are rules that simplify expressions containing powers. These properties include power multiplication, division, zero power, and negative power rules.

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\) - Multiply powers by adding exponents.
• Quotient of Powers: \(a^m / a^n = a^{m-n}\) - Divide powers by subtracting exponents.
• Power of a Power: \( (a^m)^n = a^{mn}\) - Multiply exponents when raising a power to another power.
• Zero Exponent: \(a^0 = 1\) where \(a\) is not 0, implies anything raised to the zero power is one.
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\) - Reciprocates the base for negative exponents.

These properties are applicable when simplifying radical expressions because they help identify and manage powers within expressions. For instance, in our expression simplification, recognizing how to manipulate the exponents allows for effective rationalization and simplification. Understanding these rules thoroughly empowers students to manage complex algebraic expressions effectively.