Problem 61 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive and write your answer in radical notation. $$ \sqrt[4]{r t} \cdot \sqrt[3]{r^{2} t} $$

Step-by-Step Solution

Answer

\(\sqrt[12]{r^{11} t^7}\)

Step 1: Express each radical using exponents

We begin by expressing the roots in terms of exponents. \(\sqrt[4]{r t}\) can be written as \((r t)^{1/4}\), and \(\sqrt[3]{r^{2} t}\) can be written as \((r^2 t)^{1/3}\).

Step 2: Distribute the exponents

Distribute the exponents inside the parentheses for each term. \((r t)^{1/4}\) becomes \(r^{1/4} t^{1/4}\). Similarly, \((r^2 t)^{1/3}\) becomes \(r^{2/3} t^{1/3}\).

Step 3: Multiply the expressions

Multiply the two expressions \(r^{1/4} t^{1/4}\) and \(r^{2/3} t^{1/3}\). This gives \((r^{1/4} \cdot r^{2/3})\) and \((t^{1/4} \cdot t^{1/3})\).

Step 4: Add the exponents of like bases

Since we are multiplying powers with the same base, add the exponents: \(r^{1/4 + 2/3}\) and \(t^{1/4 + 1/3}\).

Step 5: Calculate the exponents

Find a common denominator and add the exponents. For \(r\), \(1/4 + 2/3 = 3/12 + 8/12 = 11/12\). For \(t\), \(1/4 + 1/3 = 3/12 + 4/12 = 7/12\).

Step 6: Write the final expression in radical notation

Convert the exponents back into radical form: \(r^{11/12}\) and \(t^{7/12}\) can be expressed as \(\sqrt[12]{r^{11}}\cdot\sqrt[12]{t^7}\). Combine these to obtain \(\sqrt[12]{r^{11} t^7}\).

Key Concepts

Exponent Rules

Working with exponents is fundamental when simplifying expressions involving radicals and powers. Exponents represent repeated multiplication.
In expressions like \(x^a\), the base \(x\) is multiplied by itself \(a\) times. The rules for dealing with exponents make it easier to simplify complex mathematical expressions involving these powers:

• Product of Powers Rule: When multiplying two powers with the same base, you add the exponents: \(x^a \cdot x^b = x^{a+b}\).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \((x^a)^b = x^{a \cdot b}\).
• Negative Exponent Rule: A negative exponent indicates reciprocation: \(x^{-a} = 1/x^a\).

Understanding these rules helps simplify expressions, particularly when variables are involved in complex operations. In our task, adding the exponents of like bases was crucial to simplifying the expression correctly.

Radical Notation

Radical notation is a mathematical way of expressing roots, like square roots, cube roots, and so on. While radical form involves the root symbol (√), each type of root also has an equivalent expression using exponents.
For example:

• Square Root: \(\sqrt{x} = x^{1/2}\)
• Cube Root: \(\sqrt[3]{x} = x^{1/3}\)
• Fourth Root: \(\sqrt[4]{x} = x^{1/4}\)

Representing roots via exponents is particularly helpful in algebraic simplification, as it enables us to apply exponent rules. In the provided problem, converting roots to exponents allowed for clearer application of these rules, easing the multiplication and combination of the expressions. Understanding radical notation as exponents can therefore be a powerful tool for solving similar algebraic expressions.

Algebraic Simplification

Simplifying algebraic expressions is key in reducing complex mathematical equations into more manageable forms. The process often involves combining like terms and applying mathematical rules.
Here are some basic steps often used in simplification:

• Convert Terms: Change roots and complex terms to exponential form using radical notation.
• Apply Exponent Rules: Use the rules for multiplying, dividing, and combining powers with the same base.
• Combine Like Terms: Merge terms that have the same variables raised to the same power.
• Revert to Preferred Form: If needed, convert the expression back to radical form for better readability if the problem specifies it.

In the initial exercise, the expression was simplified by leveraging the exponent rules and converting between radical and exponential forms. Breaking down expressions into recognizable parts and utilizing known algebraic laws enables significant simplification, making the complicated look much simpler and manageable.