Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$\sqrt[3]{2 x}$$

Step-by-Step Solution

Answer

\((2x)^{1/3} = 2^{1/3} \cdot x^{1/3}\)

Step 1: Understand the Expression

We are given the expression \( \sqrt[3]{2x} \). This means we are trying to find a way to rewrite the cube root of \(2x\) using positive rational exponents.

Step 2: Identify the Rational Exponent Form

For any expression of the form \( \sqrt[n]{a} \), it can be rewritten using rational exponents as \( a^{1/n} \). In this problem, we have \( \sqrt[3]{2x} \), so \( n = 3 \). This implies our expression can be written as \((2x)^{1/3}\).

Step 3: Apply the Exponent to Each Term

Using the properties of exponents, \((ab)^n = a^n \cdot b^n\). Therefore, we can separate \((2x)^{1/3}\) into \(2^{1/3} \cdot x^{1/3}\).

Step 4: Combine the Results

Now, the expression \((2x)^{1/3}\) has been rewritten as \(2^{1/3} \cdot x^{1/3}\). This expression uses positive rational exponents to represent the original expression.

Key Concepts

Cube Roots

When you see the term "cube root," it means you are looking for a value that, when multiplied by itself three times, equals the original number inside the root. The cube root of a number is represented by the symbol \( \sqrt[3]{\cdot} \). Cube roots are similar to square roots but instead of finding a number that when squared equals your target, you find a number that when cubed does.In the expression \( \sqrt[3]{2x} \), we are taking the cube root of \(2x\). This means we are searching for a number that when cubed gives back \(2x\). Understanding cube roots is vital when dealing with rational exponents because they provide a foundation for expressing roots in terms of exponents.

Properties of Exponents

Exponents follow specific rules that make manipulation of expressions easier. These rules are known as the properties of exponents, and they apply across various mathematical operations. Here are some important properties:

• Product of Powers: \( a^m \times a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \times n} \)
• Power of a Product: \( (ab)^n = a^n \times b^n \)

When rewriting \( \sqrt[3]{2x} \) using exponents, we use the Power of a Product property to express it as \( 2^{1/3} \cdot x^{1/3} \). This property allows us to distribute the exponent \( \frac{1}{3} \) to each factor within the parentheses. This is a powerful tool, especially when working with variables, to simplify complex expressions.

Positive Numbers

It is crucial to provide clarity around positive numbers, especially in algebraic contexts. Positive numbers are numbers greater than zero. When you deal with expressions involving roots and exponents, it is typically assumed that variables represent positive numbers unless otherwise specified. In this exercise, it is given that the variables stand for positive numbers. This assumption is important because:

• It avoids complications like dealing with negative results from even roots, which are not real numbers.
• It simplifies the calculations because the outcomes of roots and exponentiations will naturally be positive or straightforward values.

When rewriting expressions with rational exponents, operating under the assumption of positive numbers allows for a clean transition from roots to exponent forms without worrying about undefined or complex numbers.