Problem 105 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \sqrt[3]{x}\left(\sqrt{x}-\sqrt[3]{x^{2}}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( x^{5/6} - x \).

Step 1: Express Roots as Rational Exponents

The cube root of a variable can be expressed as the variable raised to the power of one-third. Therefore, \( \sqrt[3]{x} \) can be written as \( x^{1/3} \). Similarly, \( \sqrt{x} \) is equivalent to \( x^{1/2} \) and \( \sqrt[3]{x^2} \) becomes \( x^{2/3} \).

Step 2: Rewrite the Expression

Using the rational exponents identified in Step 1, rewrite the original expression. This gives us: \[ x^{1/3} \left(x^{1/2} - x^{2/3} \right) \]

Step 3: Distribute \(x^{1/3}\) across the Parentheses

Distribute \(x^{1/3}\) into both terms inside the parentheses. This results in two separate expressions: 1. \(x^{1/3} \cdot x^{1/2}\) 2. \(x^{1/3} \cdot x^{2/3}\).

Step 4: Simplify Using Exponent Rules

Use the rule of multiplying powers which states \(a^m \cdot a^n = a^{m+n}\). Thus,- \(x^{1/3} \cdot x^{1/2} = x^{1/3 + 1/2} = x^{5/6}\) - \(x^{1/3} \cdot x^{2/3} = x^{1/3 + 2/3} = x^{1}\) Combine the terms to get the expression: \[ x^{5/6} - x \]

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, especially those involving roots and exponents, the main goal is to make the expression easier to work with. To do this, we convert roots into rational exponents and apply operations more straightforwardly. For instance, square roots or cube roots can be expressed using fractional exponents like 1/2 or 1/3. This transformation makes calculations like multiplication and division more accessible using exponent rules.

By transforming the original root expressions into rational exponents, we can simplify complex algebraic expressions. This process combines both the concepts of simplifying and using exponent rules to create a more suitable form for further mathematical operations.

Exponent Rules

Exponent rules are essential tools in algebra that help in manipulating expressions with powers. Some basic rules include:

• Product of Powers: When multiplying like bases, add the exponents, i.e., \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers: When dividing like bases, subtract the exponents, i.e., \(a^m / a^n = a^{m-n}\).
• Power of a Power: When raising a power to another power, multiply the exponents, i.e., \((a^m)^n = a^{m \cdot n}\).

In our process of simplifying, we predominantly used the product of powers rule to combine terms with the same base. Understanding and applying these rules makes algebraic expression manipulation more systematic and less prone to error.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations that together describe a mathematical reality. These expressions can include sums, differences, products, and quotients, and often require simplification to reach a form that's more workable.

Working with algebraic expressions involves understanding the relationships between the components of the expression. In our example, we simplify an expression with nested roots, which requires translating roots to rational exponents and applying operations that highlight the expression's structure. This approach helps in unraveling the complexities associated with multiple layers of operation.

Breaking down complex expressions into their simpler, exponent-based components is key to effectively managing algebraic operations. It's critical for students to grasp these concepts to solve problems accurately and efficiently.