Problem 84 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \sqrt[3]{\frac{x^{12}}{z^{6}}} $$

Step-by-Step Solution

Answer

\( \frac{x^4}{z^2} \)

Step 1: Convert the Radicals to Exponents

The expression \( \sqrt[3]{\frac{x^{12}}{z^{6}}} \) contains a cube root. We can express this as a power of \( \frac{1}{3} \). So, the expression becomes \( \left(\frac{x^{12}}{z^{6}}\right)^{\frac{1}{3}} \).

Step 2: Apply the Exponent Rule to Numerator and Denominator

Use the rule \((\frac{a}{b})^n = \frac{a^n}{b^n}\) to separate the fraction: \( \left(\frac{x^{12}}{z^{6}}\right)^{\frac{1}{3}} = \frac{x^{12 \cdot \frac{1}{3}}}{z^{6 \cdot \frac{1}{3}}} \).

Step 3: Simplify the Exponents

Calculate the exponents: \( x^{12 \cdot \frac{1}{3}} = x^4 \) and \( z^{6 \cdot \frac{1}{3}} = z^2 \).

Step 4: Write the Simplified Expression

The expression \( \frac{x^4}{z^2} \) is simplified using rational exponents.

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, we mean reducing them to their simplest form while retaining their original value and meaning. Simplification makes complex mathematical problems easier to work with. Let's understand how this works with a simple example involving a cube root.

Consider the expression \( \sqrt[3]{\frac{x^{12}}{z^{6}}} \). A cube root asks us for a number that, when multiplied by itself three times, gives the original number. This expression has the potential to look less complicated once we've used the concept of rational exponents. Notice how the simplification process goes step-by-step, breaking down the problem into more manageable pieces and transforming the expression into a more user-friendly form \( \frac{x^4}{z^2} \).

This process showcases the beauty of mathematics in turning a seemingly intricate expression into something much simpler.

Fractional Exponents

Fractional exponents can initially seem confusing, but they provide a powerful way to express roots and powers all in one notation. A fractional exponent like \( a^{\frac{m}{n}} \) can be interpreted as \( (\sqrt[n]{a})^m \). This not only represents the \( n \)-th root of \( a \) raised to the power of \( m \) but also simplifies calculations and algebraic manipulation.

In our example, we used the concept of fractional exponents by expressing the cube root \( \sqrt[3]{\frac{x^{12}}{z^{6}}} \) as \( \left(\frac{x^{12}}{z^{6}}\right)^{\frac{1}{3}} \). Here, \( \frac{1}{3} \) is the exponent representing the cube root, converting the radical into a more versatile form. This transformation is pivotal when working through more complex problems and aids in maintaining balance and harmony in mathematical equations.

Understanding and using fractional exponents effectively gives you a tool to navigate through equations with roots and powers with ease.

Algebraic Manipulation

Algebraic manipulation is a skill that allows us to rearrange, rewrite, or simplify algebraic expressions. It involves applying various algebraic rules and techniques to solve or simplify problems.

For the expression \( \left(\frac{x^{12}}{z^{6}}\right)^{\frac{1}{3}} \), we employ the property \((\frac{a}{b})^n = \frac{a^n}{b^n}\). Using this rule, we can separate the fraction into its numerator and denominator, raising each part separately to the power of \( \frac{1}{3} \). This gives us \( \frac{x^{12 \cdot \frac{1}{3}}}{z^{6 \cdot \frac{1}{3}}} \) which then simplifies to \( \frac{x^4}{z^2} \).

By consistently applying these algebraic principles, we can transform and simplify expressions. Discovering paths to simplification unveils mathematical relationships, often leading to clearer insights and solutions.