Problem 23 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt[3]{\frac{2 x}{27 y^{12}}} $$

Step-by-Step Solution

Answer

\( \frac{2^{1/3}x^{1/3}}{3y^{4}} \)

Step 1: Express the Root as a Power

Rewrite the cube root expression using rational exponents. The expression \( \sqrt[3]{\frac{2x}{27y^{12}}} \) becomes \( \left(\frac{2x}{27y^{12}}\right)^{1/3} \).

Step 2: Apply the Power to Numerator and Denominator

Apply the exponent \( \frac{1}{3} \) to both the numerator and the denominator of the fraction inside the parentheses: \( \frac{(2x)^{1/3}}{(27y^{12})^{1/3}} \).

Step 3: Simplify the Numerator

Simplify \( (2x)^{1/3} \). This gets broken down to \( 2^{1/3}x^{1/3} \).

Step 4: Simplify the Denominator - Constant

Simplify \( 27^{1/3} \). Since 27 is a perfect cube, \( 27^{1/3} = 3 \).

Step 5: Simplify the Denominator - Variable

Simplify \( (y^{12})^{1/3} \). Use the power of a power rule: multiply the exponents, resulting in \( y^{4} \).

Step 6: Combine Simplified Components

Combine the simplified parts to get the final expression: \( \frac{2^{1/3}x^{1/3}}{3y^{4}} \).

Key Concepts

Rational Exponents

Rational exponents are a way of expressing roots using exponents or powers. The exponent is written as a fraction, where the numerator represents the power, and the denominator represents the root. For example, the cube root of a variable, which we typically write as \( \sqrt[3]{a} \), can be expressed using a rational exponent as \( a^{1/3} \). Rational exponents make it easier to perform algebraic operations involving roots, like multiplication and division, since they follow the same rules as integer exponents.

• They allow us to represent roots and powers within a single framework.
• They simplify the process of multiplying and dividing roots and powers.

Using rational exponents transforms roots into a familiar exponential form, making expressions easier to manipulate and simplify.

Simplifying Expressions

Simplifying expressions involves rewriting them in the simplest form possible, making it easier to understand and work with. This process uses algebraic rules and properties, such as the power rule, quotient rule, and rational exponents.
When simplifying a fractional expression with an exponent, like in the initial problem, break each component down separately. First, apply the exponent to both the numerator and the denominator, then simplify each part.
This method ensures clarity and precision. For instance, applying \( \frac{1}{3} \) to both the numerator \( 2x \) and the denominator \( 27y^{12} \) simplifies to \( 2^{1/3}x^{1/3} \) and \( 3y^4 \) respectively.

• Always start by expressing roots as powers when simplifying.
• Break down each component of the expression individually.
• Combine the simplified components to achieve the final simplified form.

These steps help to reach the most reduced form of an expression, enabling easier computation and analysis.

Power Rule for Exponents

The power rule for exponents is a fundamental principle in algebra that simplifies expressions with exponents. It states that when you raise a power to another power, you multiply the exponents.
For instance, \( (x^m)^n = x^{mn} \). This rule is extremely useful when dealing with complex expressions, especially those with nested exponents.

• It's used to simplify expressions like \( (y^{12})^{1/3} \), resulting in \( y^4 \) by multiplying the exponents, 12 and \( \frac{1}{3} \).
• It provides a straightforward method to handle expressions with multiple layers of exponents.

This rule also complements other laws of exponents, allowing for a more systematic approach to handling mathematical expressions, especially those involving rational exponents. Understanding the power rule is crucial for simplifying and solving many algebraic problems effectively.