Problem 100 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \sqrt[3]{t} \cdot \sqrt[5]{t} $$

Step-by-Step Solution

Answer

The simplified expression is \(t^{8/15}\).

Step 1: Express Each Root with Rational Exponents

First, convert the radical forms into expressions with rational exponents. The cube root of a number \(t\) can be written as \(t^{1/3}\). Similarly, the fifth root of \(t\) can be expressed as \(t^{1/5}\). So, the expression \(\sqrt[3]{t} \cdot \sqrt[5]{t}\) becomes \(t^{1/3} \cdot t^{1/5}\).

Step 2: Apply the Laws of Exponents

Using the rule that \(a^m \cdot a^n = a^{m+n}\), we add the exponents of \(t\). This gives us \(t^{1/3 + 1/5}\).

Step 3: Simplify the Exponents

Add the exponents: \(\frac{1}{3} + \frac{1}{5}\). To do this, find a common denominator. The least common denominator of 3 and 5 is 15. Convert the fractions: \(\frac{1}{3} = \frac{5}{15}\) and \(\frac{1}{5} = \frac{3}{15}\). So, the sum is \(\frac{5}{15} + \frac{3}{15} = \frac{8}{15}\).

Step 4: Write the Final Simplified Expression

The simplified expression is \(t^{8/15}\), using rational exponents.

Key Concepts

Laws of Exponents

The laws of exponents are fundamental rules that simplify working with expressions that involve powers. These laws help us understand how to handle and manipulate expressions using the properties of exponents. Here are a few key rules:

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

These laws allow us to transform complex expressions into easier expressions, making calculations more manageable. Applying these laws requires understanding how exponents interact when they share the same base. This comes in handy when trying to simplify expressions involving rational exponents or roots.

Simplifying Expressions

Simplifying expressions is a critical skill in algebra that allows us to rewrite expressions in a more efficient or readable form. The main goal is to make the expression as simple as possible. This involves a few key steps:

• Combine like terms: This might mean terms that have the same exponent, as seen in our exercise, where \(t^{1/3} \cdot t^{1/5}\) was simplified using the product of powers rule.
• Simplify fractions: Simplifying the fractional part of an exponent, like in \(\frac{1}{3} + \frac{1}{5}\), leads to a single rational exponent \(\frac{8}{15}\).

Simplifying is about reducing expressions to their simplest form while preserving equivalence. This process can involve rationalizing denominators or reducing exponents, which also plays into simplifying complex expressions into clear, concise results.

Common Denominator

A common denominator is necessary when adding or subtracting fractions. It's the same denominator for each fraction involved, allowing you to combine them. Here’s how you typically find it:

• Identify each denominator: Look at each part of the fractions, for example, \(\frac{1}{3}\) and \(\frac{1}{5}\).
• Find the least common multiple (LCM): The LCM is the smallest number that each denominator divides into. For \(3\) and \(5\), the LCM is \(15\).
• Adjust each fraction to have this common denominator: Convert each fraction using the LCM, making sure the numerator adjusts accordingly. For instance, \(\frac{1}{3} = \frac{5}{15}\) and \(\frac{1}{5} = \frac{3}{15}\).

Common denominators are crucial when combining fractions, as they align terms under a single denominator, allowing straightforward arithmetic. This concept is often employed in algebra when working with rational expressions or simplifying exponents.