Problem 50 - Algebra for College Students

Question

Write each of the following using positive rational exponents. For example, \(\sqrt{a b}=(a b)^{\frac{1}{2}}=a^{\frac{1}{2}} b^{\frac{1}{2}}\). \(\sqrt[5]{x^{2} y^{4}}\)

Step-by-Step Solution

Answer

\(\sqrt[5]{x^{2} y^{4}} = x^{\frac{2}{5}} y^{\frac{4}{5}}\).

Step 1: Understand the Radical

The given expression is a radical, specifically a fifth root: \(\sqrt[5]{x^2 y^4}\). The index of the radical is 5, indicating a fifth root, and the radicand is \(x^2 y^4\).

Step 2: Write the Radical as an Exponent

To convert from a radical to an expression with exponents, use the rule \(\sqrt[n]{a} = a^{\frac{1}{n}}\). Therefore, \(\sqrt[5]{x^2 y^4}\) becomes \((x^2 y^4)^{\frac{1}{5}}\).

Step 3: Apply the Power to Each Factor

Distribute the fractional exponent to each factor in the expression. This means applying the exponent \(\frac{1}{5}\) to both \(x^2\) and \(y^4\).

Step 4: Simplify Each Factor with the Exponent

Each factor within the parentheses is raised to the power of \(\frac{1}{5}\). Thus, \((x^2)^{\frac{1}{5}} = x^{\frac{2}{5}}\) and \((y^4)^{\frac{1}{5}} = y^{\frac{4}{5}}\).

Step 5: Write the Final Expression

Combine the simplified factors to form the final expression in terms of positive rational exponents. Therefore, the expression becomes \(x^{\frac{2}{5}} y^{\frac{4}{5}}\).

Key Concepts

Radicals

Radicals are an essential part of mathematics that express roots, like square roots or cube roots. They are identified by the radical symbol \( \sqrt{} \). One key component of a radical is its index, which indicates the degree of the root. For instance, a radical symbol without a visible index is a square root. However, when it has a small numeral, it tells us which root is being taken.

• Square roots: The common radical \( \sqrt{a} \) entails a square root.
• Cubic and higher roots: Radicals with indices such as 3, 4, 5, etc., denote cube roots, fourth roots, and so on, like \( \sqrt[3]{a} \) or \( \sqrt[5]{a} \).

Understanding radicals involves recognizing the radicand, the number underneath the radical sign, and knowing how to change between radical forms and exponents. This becomes crucial, especially in problem-solving where expressions may be simplified using rational exponents to make calculations easier.

Exponent Rules

Exponent rules help simplify expressions involving powers and are essential for working with rational exponents. When radicals are rewritten as exponents, using these rules makes calculations more straightforward.Whenever you see an expression like \(a^{m/n}\), it combines a power and a root:

• Power Rule: If you multiply exponents with the same base, you add the exponents. Example: \(a^m \times a^n = a^{m+n}\).
• Power of a Power Rule: When taking a power of a power, multiply the exponents. For example, \((a^m)^n = a^{mn}\).
• Product to Power Rule: When raising a product to a power, apply the exponent to each factor. For example, \((ab)^n = a^n b^n\).
• Rational Exponents: The expression \(a^{m/n}\) means take the \(n\)-th root of \(a^m\). This is especially useful in converting between radical notation and exponential form.

Mathematical Expressions

Mathematical expressions use symbols and numbers to represent quantities or relationships. Expressions can include numbers, variables, operators (like plus and minus), and even functions.In the context of radicals and exponent rules, expressions involving rational exponents, like \(x^{\frac{2}{5}}y^{\frac{4}{5}}\), show a powerful way to represent roots and powers in a single format. These expressions allow for efficient simplification and manipulation.

• Combining Terms: When simplifying expressions, like \(x^{\frac{2}{5}}y^{\frac{4}{5}}\), it's crucial to identify and combine like terms properly.
• Order of Operations: Remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) when evaluating expressions.
• Simplification: Before solving equations, ensure expressions are in their simplest form using the laws of exponents and radicals accordingly.

Understanding mathematical expressions empowers students to model and solve real-world problems effectively, thanks to their versatility and scope.