Problem 95 - Intermediate Algebra

Question

Use rational expressions to write as a single radical expression. $$ \sqrt{5 r} \cdot \sqrt[3]{s} $$

Step-by-Step Solution

Answer

Combine under a 6th root: \(\sqrt[6]{(5r)^3 s^2}\).

Step 1: Express Radicals with Rational Exponents

First, rewrite each radical expression using rational exponents. The square root \( \sqrt{5r} \) can be expressed as \( (5r)^{\frac{1}{2}} \), and the cube root \( \sqrt[3]{s} \) as \( s^{\frac{1}{3}} \).

Step 2: Combine the Expressions Using Multiplication

Next, multiply the two rational exponent expressions: \((5r)^{\frac{1}{2}} \cdot s^{\frac{1}{3}}\). According to the properties of exponents, this can be expressed as a single expression: \((5r)^{\frac{1}{2}} s^{\frac{1}{3}}\).

Step 3: Convert Back to a Single Radical Expression

Combine the rational exponents into a single radical expression. The expressions can be simplified into one radical using a common radical (root) form. Since the radicals were of order 2 and 3, find a common multiple, which is the 6th root: \(\sqrt[6]{(5r)^3 s^2}\), reducing the rational exponents to this common root.

Key Concepts

Rational Exponents

Rational exponents allow us to express roots and powers in a unified way, making complex problems simpler to solve. When you encounter a radical, like a square root or cube root, it can be rewritten as a rational exponent. This means any root can be represented as a fraction. For example:

• The square root \( \sqrt{x} \) is the same as \( x^{\frac{1}{2}} \).
• The cube root \( \sqrt[3]{x} \) is \( x^{\frac{1}{3}} \).
• In general, the \( n \)-th root of \( x \) is \( x^{\frac{1}{n}} \).

This conversion helps in multiplying and dividing roots by applying the well-known rules of exponents, simplifying calculation processes. By using rational exponents, you can treat a root as a power, thus integrating it seamlessly with other powers when performing operations.

Radical Expressions

Radical expressions involve roots, whether they are square roots, cube roots, or any other kind of roots. The expression \( \sqrt{5r} \cdot \sqrt[3]{s} \) includes both a square root and a cube root. Such expressions can be rewritten and worked on using rational exponents, making them easier to manage and combine. When converting radicals to rational exponents, you're translating the operations and logic into a more consistent mathematical framework. This helps especially when you need to combine different types of radicals into a single expression. For instance, combining \( \sqrt{5r} \) and \( \sqrt[3]{s} \) as a single radical involves transforming both into a consistent format using exponents first, then back to a radical form, which simplifies processes where multiple types of roots are involved. Dealing with radical expressions gets easier as you get familiar with converting between radical and rational forms, streamlining how you solve equations and inequalities that feature these elements.

Properties of Exponents

The properties of exponents are powerful tools in simplifying expressions, especially when dealing with rational exponents and radicals. Here are some important properties to remember:

• Product of Powers: When you multiply two expressions with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power: Raising a power to another power means you multiply the exponents: \( (a^m)^n = a^{m\cdot n} \).
• Power of a Product: Raising a whole product to an exponent means each factor is raised to that exponent: \( (ab)^m = a^m b^m \).

These properties allow us to manipulate expressions to find alternate forms or simplify complex expressions. When working on an example like \( \sqrt{5r} \cdot \sqrt[3]{s} \), after expressing them as rational exponents like \( (5r)^{\frac{1}{2}} \) and \( s^{\frac{1}{3}} \), you can multiply them using these properties.Finally, rational exponents use these properties to move back and forth between radical and exponent forms, combining into single expressions where appropriate.