Problem 48 - Precalculus: Functions and Graphs

Question

Rewrite the expression using rational exponents. $$\sqrt[3]{x^{5}}$$

Step-by-Step Solution

Answer

The expression \( \sqrt[3]{x^5} \) is rewritten as \( x^{5/3} \) using rational exponents.

Step 1: Understanding the Problem

We need to rewrite the expression \( \sqrt[3]{x^5} \) using rational exponents instead of a radical. A radical expression like this can also be expressed using an exponent.

Step 2: Rewriting the Expression

To convert the radical expression \( \sqrt[3]{x^5} \) to a rational exponent, we use the formula \( \sqrt[n]{x^m} = x^{m/n} \). Here, \( n = 3 \) and \( m = 5 \).

Step 3: Applying the Formula

Using the formula, we rewrite \( \sqrt[3]{x^5} \) as \( x^{5/3} \). This turns the cube root and power into a single exponent.

Step 4: Simplifying the Expression

The expression \( x^{5/3} \) is already simplified as a rational exponent. Ensure it is presented in reduced terms.

Key Concepts

Cube Roots

A cube root is a special type of root that is used when you need to find a number which, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \). The cube root of a number \( x \) is represented as \( \sqrt[3]{x} \). It's different from a square root, which involves a factor of two. Understanding cubic roots is essential as they're used in various mathematical concepts and problems, including the use of rational exponents.

• The notation \( \sqrt[3]{x} \) indicates a cube root.
• Cube roots help simplify expressions involving the third power of numbers.
• They're useful in real-world applications like calculating volume in geometry.

Exponents

Exponents represent how many times a number, called the base, is multiplied by itself. For instance, with the expression \( x^n \), \( x \) is the base and \( n \) is the exponent, denoting that \( x \) should be used as a factor \( n \) times. Exponents play a significant role in simplifying and solving mathematical expressions.
Using exponents effectively can transform radical expressions, like cube roots, into a different format. This makes the calculation easier to understand and manage.

• An exponent of 2 implies squaring the base (e.g., \( x^2 \))
• An exponent of 3 implies cubing the base (e.g., \( x^3 \))
• Negative exponents indicate reciprocal (e.g., \( x^{-n} = \frac{1}{x^n} \))

Radical Expressions

Radical expressions are mathematical expressions that include a radical symbol, which is used to depict roots of numbers. The most common radical is the square root, \( \sqrt{} \), but others like the cube root, \( \sqrt[3]{} \), and higher roots also belong to this category.
Rewriting radical expressions with rational exponents can simplify mathematical operations, allowing easier manipulation and calculation. For example, converting \( \sqrt[3]{x^5} \) to \( x^{5/3} \) lets us handle the expression as a simple power of \( x \).

• Radical expressions can often be rewritten for simplicity using rational exponents.
• They appear frequently in algebra, calculus, and geometry.
• Conversion between radical and exponential forms requires understanding both notations.