Problem 21 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The expression is \(2^{\frac{1}{3}} \cdot x^{\frac{1}{3}}\).

Step 1: Understand Cube Root Notation

The expression \( \sqrt[3]{2x} \) represents the cube root of \( 2x \). A cube root can be rewritten using an exponent of \( \frac{1}{3} \). This is based on the property that \( \sqrt[n]{a} = a^{\frac{1}{n}} \).

Step 2: Apply Rational Exponent Rule

Using the rule from Step 1, rewrite \( \sqrt[3]{2x} \) as \( (2x)^{\frac{1}{3}} \). This is due to the fact that taking the cube root of a number is equivalent to raising it to the power of \( \frac{1}{3} \).

Step 3: Distribute the Exponent

Apply the exponent to both the number and the variable separately. The expression \((2x)^{\frac{1}{3}}\) is equivalent to \(2^{\frac{1}{3}} \cdot x^{\frac{1}{3}}\). This invokes the property \( (a \, b)^n = a^n \, b^n \).

Key Concepts

Cube Root

The concept of a cube root is fundamental in mathematics. The expression \( \sqrt[3]{a} \) is read as "the cube root of \( a \)". Just like how a square root "undoes" squaring a number, a cube root "unwinds" cubing a number. For example, if \( b^3 = a \), then \( \sqrt[3]{a} = b \).
Simply put, cube root of a number \( a \) is the number that, when cubed (or multiplied by itself two more times), results in \( a \). A neat way to handle cube roots in algebra is by expressing them using exponents. Hence, \( \sqrt[3]{a} \) can also be written as \( a^{\frac{1}{3}} \).

• The cube root notation visually indicates a different mathematical operation compared to square roots.
• Understanding cube roots as rational exponents makes many algebraic manipulations more straightforward.

Exponent Rules

Exponent rules allow us to manipulate expressions involving powers easily. When dealing with rational exponents, these rules still hold true. For example:

• Product of Powers Rule: \( a^m \cdot a^n = a^{m+n} \) allows us to combine powers when multiplying like bases.
• Quotient of Powers Rule: \( \frac{a^m}{a^n} = a^{m-n} \) lets us simplify expressions by subtracting exponents during division.
• Power of a Power Rule: \( (a^m)^n = a^{mn} \) means we multiply exponents when raising a power to another power.

When you write an expression using rational exponents like \((2x)^{\frac{1}{3}}\), consider treating it as \((2^{\frac{1}{3}}) \cdot (x^{\frac{1}{3}})\). This way, exponents distribute over multiplication, following the property \((a \, b)^n = a^n \, b^n\). It's a powerful rule that simplifies algebraic expressions with ease.

Positive Numbers

In algebra, we often assume variables represent positive numbers unless stated otherwise. Working with positive numbers has its benefits:

• Certain operations, like exponentiation, can give different results based on the sign of the number. Positive numbers provide consistency.
• Positive numbers ensure that expressions like roots and logarithms stay within the domain of real numbers. This is crucial for cube roots to avoid any complexities of imaginary numbers.

When dealing with rational exponents, assuming positivity ensures that expressions remain valid and straightforward. For example, for \( x > 0 \), \( x^{\frac{1}{3}} \) is always defined and stays within the realm of real numbers. This assumption helps streamline proofs and simplifies understanding in mathematics.
Essentially, positivity implies a wider applicability of mathematical operations and keeps results in the real number system, aiding in more intuitive and logical reasoning.