Problem 21 - College Algebra with Modeling and Visualization

Question

Use positive exponents to rewrite. $$ \sqrt[3]{z^{5}} $$

Step-by-Step Solution

Answer

The expression is rewritten as \( z^{\frac{5}{3}} \).

Step 1: Understand the Radical Expression

The expression \( \sqrt[3]{z^{5}} \) is a cube root of \( z^5 \). To solve this, we need to rewrite the radical expression using exponents.

Step 2: Apply the Fractional Exponent Rule

The expression \( \sqrt[3]{z^5} \) can be rewritten as an expression with a fractional exponent. The cube root is equivalent to raising to the power of \( \frac{1}{3} \). Thus, \( \sqrt[3]{z^5} = z^{5 \cdot \frac{1}{3}} \).

Step 3: Simplify the Exponent

Multiply the exponent \( 5 \) by \( \frac{1}{3} \) to simplify: \( 5 \times \frac{1}{3} = \frac{5}{3} \). Thus, the expression becomes \( z^{\frac{5}{3}} \).

Step 4: Conclusion

The initial expression \( \sqrt[3]{z^5} \) is rewritten as \( z^{\frac{5}{3}} \) with a positive exponent.

Key Concepts

Radical Expressions

When you see a radical expression, you'll often find a root symbol—like a square root or a cube root—that represents the operation of taking a root of a number or a variable. For example, in the expression \( \sqrt[3]{z^5} \), the symbol \( \sqrt[3]{} \) stands for the cube root. The number under the root sign is called the radicand, which in this case is \( z^5 \).

Radicals are used to represent root operations very compactly. However, to simplify or perform operations with radicals, it’s often easier to convert them to expressions with exponents. This can transform complex radical expressions into more manageable forms.

Working with radicals is a foundational skill in algebra as they frequently appear in problems solving scenarios and require a solid understanding for manipulation in equations.

Fractional Exponents

Fractional exponents stem from the relationship between powers and roots, making them incredibly useful in expressing radical expressions otherwise.

In mathematics, taking the root of a number can be expressed using exponents. Specifically, the \( n \)-th root of a number is expressed as raising the number to the power of \( \frac{1}{n} \). This ties directly to our example,

• \( \sqrt[3]{z^5} \) is equivalent to \( z^{5 \times \frac{1}{3}} \) because a cube root is the same as raising something to the power of \( \frac{1}{3} \).
• Here, \( 5 \times \frac{1}{3} \) equals \( \frac{5}{3} \), creating the expression \( z^{\frac{5}{3}} \).

This transformation from radicals to fractional exponents helps in simplifying and analyzing expressions, making it a vital part of algebraic manipulation.

Simplifying Expressions

Simplifying expressions involves rewriting them in the most efficient form while maintaining their equivalence. This process often reduces complexity and makes further computations more straightforward. When dealing with exponents, simplification can involve combining like terms, modifying powers, or converting radicals to fractional exponents to streamline expressions.

In the example \( z^{\frac{5}{3}} \), the expression is reconstructed into a simple and concise format from its initial radical form \( \sqrt[3]{z^5} \). By converting to fractional exponents, we achieved a more workable expression.

Understanding how to simplify expressions efficiently is crucial for advanced math problem solving. It aids in making sense of equations and expressions quickly and is an invaluable skill when dealing with complex algebraic tasks.