Problem 39 - Algebra for College Students

Question

Write each of the following in radical form. For example, \(3 x^{\frac{2}{3}}=3 \sqrt[3]{x^{2}}\). \((2 a-3 b)^{\frac{2}{3}}\)

Step-by-Step Solution

Answer

The expression is \( \sqrt[3]{(2a - 3b)^2} \).

Step 1: Identify Exponent Structure

The given expression is \( (2a - 3b)^{\frac{2}{3}} \). This is in the form \( x^{\frac{m}{n}} \), where \( m = 2\) and \( n = 3\). The expression is a fractional exponent.

Step 2: Apply Radical Form Transformation

Convert the expression \( (2a - 3b)^{\frac{2}{3}} \) to radical form using the rule \( x^{\frac{m}{n}} = \sqrt[n]{x^m} \). Here, \( (2a - 3b)^{\frac{2}{3}} = \sqrt[3]{(2a - 3b)^2} \).

Step 3: Simplify Final Expression

Express the final solution, which is \( \sqrt[3]{(2a - 3b)^2} \), and ensure all elements are correctly placed under the radical sign.

Key Concepts

Fractional Exponents

Fractional exponents might look intimidating, but they aren't as tricky as they appear. In essence, a fractional exponent indicates both a power and a root. The numerator of the fraction tells us the power, while the denominator indicates the root. For example, in the fractional exponent \( x^{\frac{m}{n}} \), \( m \) represents the power, and \( n \) signifies the root.

Understanding fractional exponents is crucial because they are prevalent in various branches of mathematics and science. When you encounter expressions like \( (2a - 3b)^{\frac{2}{3}} \), you should interpret it as the base \( (2a - 3b) \) being raised to the power of 2 and then taking the cube root. Here's how it works:

• The numerator (2) indicates we square the expression \((2a - 3b)\).
• The denominator (3) suggests we take the cube root of the result.

When tackled step-by-step, fractional exponents provide a straightforward way to handle powers and roots in a single expression.

Radical Form Transformation

Converting expressions with fractional exponents into radical form is like translating a math sentence into another language. This transformation helps clarify the operations involved. The general rule for this transformation is: \( x^{\frac{m}{n}} = \sqrt[n]{x^m} \).

By applying this rule, you can rewrite the fractional exponent in a more familiar radical form. Let's consider the expression \((2a - 3b)^{\frac{2}{3}}\):

• According to the rule, write \( (2a - 3b)^{\frac{2}{3}} \) as \(\sqrt[3]{(2a - 3b)^2}\).
• This means you first square \((2a - 3b)\) and then take the cube root of the result.

This transformation can greatly simplify complex expressions by making the root operation explicit, which can be extremely useful in further calculations or problem-solving scenarios.

Exponent Rules

Understanding and applying exponent rules is essential for manipulating expressions and solving equations. Exponent rules help you simplify expressions by providing a set of guidelines on how to handle powers. Here are some critical rules:

• Product of Powers Rule: \( x^a \times x^b = x^{a+b} \).
• Power of a Power Rule: \((x^a)^b = x^{a \times b}\).
• Power of a Product Rule: \((xy)^a = x^a \times y^a\).
• Negative Exponent Rule: \( x^{-a} = \frac{1}{x^a} \).

When working with fractional exponents, the power of a power rule is particularly useful. For instance, in \( (2a - 3b)^{\frac{2}{3}} \), you're essentially applying a power of 2 and a root of 3. Understanding these rules helps in rationalizing expressions and solving algebraic equations efficiently.