Problem 23 - A Graphical Approach to Precalculus with Limits

Question

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$\sqrt[3]{z^{5}}$$

Step-by-Step Solution

Answer

\( z^{\frac{5}{3}} \)

Step 1: Identify the Expression

The given expression is \( \sqrt[3]{z^5} \). This is a radical expression with a cube root.

Step 2: Convert the Radical to an Exponent

The cube root of any number or variable can be written as an exponent of \( \frac{1}{3} \). Therefore, \( \sqrt[3]{z^5} \) can be rewritten as \( (z^5)^{\frac{1}{3}} \).

Step 3: Apply the Power Rule of Exponents

Use the power rule \((a^m)^n = a^{m \cdot n}\) to simplify \((z^5)^{\frac{1}{3}}\). Multiplying the exponents gives \( z^{5 \cdot \frac{1}{3}} \).

Step 4: Simplify the Exponent

Multiply the exponents: \( 5 \cdot \frac{1}{3} = \frac{5}{3} \). Thus, \((z^5)^{\frac{1}{3}} = z^{\frac{5}{3}}\).

Step 5: Rewrite the Expression

The original expression \( \sqrt[3]{z^5} \) is correctly rewritten with positive rational exponents as \( z^{\frac{5}{3}} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, which are essential to simplifying expressions in math. They help indicate the root of a number, like finding the number that multiplies by itself a certain number of times to yield a given value.

In the expression \( \sqrt[3]{z^5} \), "\(\sqrt[3]{}\)" represents the cube root. Radical notation is an alternative way to express fractional exponents.

Understanding radicals means recognizing how they simplify expressions when rewritten with exponents. By expressing radicals as exponents, you unlock the flexibility to use exponent rules, making calculations easier and more manageable.

Exponent Rules

Exponent rules provide the foundation for simplifying expressions involving powers. These rules help to navigate through calculations and include several key properties:

• Product of Powers: When multiplying like bases, add their exponents: \( a^m \cdot a^n = a^{m+n} \).
• Power of a Power: When taking a power of a power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \). This rule is essential when converting radical expressions to rational exponent form.
• Power of a Product: Distribute the exponent to each factor in the product: \( (ab)^n = a^n \cdot b^n \).

Applying these rules lets us transform expressions like \( (z^5)^{\frac{1}{3}} \) into \( z^{\frac{5}{3}} \), simplifying computations and clarifying the relationships in algebraic equations.

Cube Root

The cube root of a number is one of the fundamental radical operations in mathematics. It is the value that, when used three times in multiplication, gives the original number. For example, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \).

When working with variables, such as in \( \sqrt[3]{z^5} \), we can express the cube root as an exponent. This is done by using the fractional notation \( z^{\frac{1}{3}} \). By writing the cube root in this way, it becomes straightforward to apply the exponent rules, simplifying complex expressions easily.

Understanding cube roots in this form not only simplifies calculations but also provides deeper insights into equations where variables are involved. It allows students to move confidently between different representations and find fresh insights through manipulating algebraic expressions.